Dominic Raab said much would rest on the phone call held between the prime minister and European Commission President Ursula von der Leyen when they speak at 11am.
Talks between the UK and the EU began again at 8am but have been deadlocked for days over disagreements on fishing rights and the so-called level playing field “ratchet” that would effectively tie the UK to future EU standards.
Asked on BBC One’s The Andrew Marr Show if talks could continue past Sunday, Mr Raab said although “finality” is needed he would “never say never”.
“The bar is quite high for us to continue to keep talking, we would need at a political level a commitment to move on those two key issues,” Mr Raab said.
Speaking on Sky News's Sophy Ridge on Sunday, Mr Raab said there is "still a long way to go" but insisted the government does "want to get a deal if it's at all possible".
Talks are continuing on what has been billed as the final day, with the probability of a no-deal outcome looking more likely with government sources saying the EU's offer "remains unacceptable".
Sources in the British government warned the offer on the table from the EU “remains unacceptable” to the UK, with the country teetering on the edge of a no-deal Brexit that is predicted to cost jobs and force food prices to increase.
Brexit trade talks - the sticking points at a glance:
- Fishing rights: The UK wants total control over its own fishing waters after the Brexit transition period ends, with a 12 mile exclusion zone around the British Isles banning all foreign vessels. The EU wants the UK to stick to the Common Fisheries Policy, an EU agreement which gives member nations the rights to fish in European waters - more here.
- Level Playing Field: This is a concept all EU nations agree to, which ensures member nations cannot undercut others by setting their own rules on issues such as the environment, taxation and state aid. The EU says a zero-tariff trade deal is dependent on the UK agreeing to a level playing field. The UK disagrees, saying a fundamental aspect of Brexit is that the UK will be able to set its own rules.
- Governance of a deal: It's likely that any trade deal will eventually result in disputes. The EU wants the European Court of Justice to be the final authority in ruling over disputes. The UK says the ECJ should have no role and final decisions should be made by a bespoke arbiter. But the statements from both sides suggested that while further discussions would be held, substantial movement on the key issues had not been made.
Speaking on Sophy Ridge on Sunday, Mr Raab appeared to point the finger of blame at the EU over a lack of deal being done, saying the bloc has "got to come to terms" with letting the UK gain independence.
"We've gone into this with a spirit of pragmatism, of optimism, good will and what we've just found in the last analysis and at the last stage the EU has found letting go of its control over the UK very difficult - I think emotionally and politically if you like, they've got to come to terms with that," Mr Raab said.
He continued that the UK had worked "very hard" at a technical level during the recent negotiations in Brussels but argued that there needed to be political "willing" to secure a Brexit trade deal.
"We want to be treated like any other independent self-respecting democracy," Mr Raab said.
"If the EU can accept that at a political level then there's every reason to be confident, but there is still I think a long way to go."
He highlighted "the issue of fisheries and the issue of the so-called level playing field" as two sticking points in negotiations and warned that French fishing boats will have "zero access guaranteed" to the UK's waters if there is no trade deal.
Asked about navy ships patrolling the waters, he said: "The bottom line is actually if we do leave on WTO [World Trade Organization] terms we'll be an independent coastal state. Of course we're going to enforce our waters around fisheries and whatever else.
"And of course for the French and others, that will mean - you know, forget those outlandish terms that they were asking of us - their fishing industries would have zero access guaranteed."
There does appear to be willingness from the EU for negotiations to continue.
Ireland's prime minister said dialogue should continue as long as there is the possibility of a post-Brexit trade deal.
Micheal Martin said 97% of issues had already been agreed between Britain and the EU.
He told BBC One's The Andrew Marr Show: "I think it would be an appalling failure of statecraft if we were not in a position to get a deal over the line."
He added: "It is vital that we do not have an acrimonious break-up because that would be very damaging...
"Where the dialogue continues, that gives me hope," he said.
While Spain's Foreign Minister Arancha Gonzalez said a no-deal outcome should be avoided "at all costs".
She told Sky News' Sophy Ridge On Sunday: "No-deal in the current circumstances would be extremely negative for our economies.
"And if you go by what economists are saying, and there is plenty of literature on that, the UK would suffer even more than the European Union.
"We both will suffer, more on the UK side, which I think is something we should try to avoid at all costs."
Mr Johnson and his European counterpart had a dinner meeting on Wednesday in Brussels after several days of talks which appeared to have hit a stalemate.
When the pair met, they both agreed a firm decision on the future of negotiations was needed by the end of the weekend.
However, speaking on Saturday's News At Ten, ITV News Political Correspondent Daniel Hewitt said that Sunday's deadline is "self-imposed" and the two sides could decide to give themselves more time.
However, a decision must be made by the end of the year, if one if not, the UK will trade with the EU on World Trade Organization rules and tariffs.
The outlook after discussions on Saturday was described as “very difficult” but officials said the prime minister was determined to explore every option to secure a free trade agreement.
A government source said: “Talks are continuing overnight, but as things stand the offer on the table from the EU remains unacceptable.
“The Prime Minister will leave no stone unturned in this process, but he is absolutely clear: any agreement must be fair and respect the fundamental position that the UK will be a sovereign nation in three weeks’ time.”
Mr Johnson is expected to give a press conference or issue a recorded statement to update the nation once he finishes a call with Europe’s top official, but no timing has yet been given.
The Conservative Party leader and Ms von der Leyen have warned a no-deal outcome looks more likely than an agreement in the trade negotiations.
With the prime minister describing no-deal as “very, very likely”, the government has stepped up preparations for leaving the single market when transition arrangements end on December 31, with Mr Johnson taking personal control of ensuring the country is ready.
He is leading a “Super XO” committee to oversee preparations as ministers look to ensure food, medicines – including coronavirus vaccines – and other critical goods can continue to reach the country uninterrupted next year.
The Confederation of British Industry (CBI) has called for “grace periods” to be agreed so businesses do not face a “damaging cliff edge” if there proves to be no meeting of minds on a deal.
In a move likely to incense EU leaders, a government spokesperson revealed the UK had “run live exercises” that involved scrambling “naval vessels to respond to threats of illegal fishing in our soon-to-be sovereign waters” as part of readiness efforts.
It follows confirmation from the Ministry of Defence that four Royal Navy gunboats have been placed on standby to guard British waters from EU trawlers if there is no agreement – an announcement that has been greeted with anger by some senior Tories.
Reports also suggest ministers are considering beefing-up Navy powers in legislation to authorise them to board and arrest people fishing in contravention of post-Brexit rules.
Tobias Ellwood, Conservative chairman of the Commons Defence Committee, called the threat “irresponsible” and warned it would damage Britain’s international reputation.
'It's sad we're preparing to square up to a Nato ally': Conservative MP Tobias Ellwood critical of UK stance
Military officials disagreed, however, with Admiral Lord West, a former chief of naval staff, arguing that fiery past clashes between fishermen in the Channel suggested armed forces intervention could be required.
French MEP Pierre Karleskind, chair of the European Parliament’s Committee on Fisheries, called for “calm” heads when it came to working on a fisheries solution if the talks collapsed.
Brussels has called for the status quo on fishing rights to continue for 12 months in the event of no-deal – a request that appears to have been rejected after the government revealed plans to scale up patrols.
Mr Johnson, in a speech at a climate change summit on Saturday, appeared to take a dig at French President Emmanuel Macron over the fishing row.
Mr Macron is said to have threatened to veto a UK-EU deal after expressing dissatisfaction at the new quota terms being thrashed out for French fishermen.
In his closing remarks, the prime minister thanked summit co-host Mr Macron, adding that he knew the En Marche! leader “shares my keen interest in protecting the ecosystems of our seas”.
Tory MP David Jones wrote in the Sunday Telegraph the PM "must keep faith with the British people and resist any temptation to accept a sub-optimal deal that would cheat them of the sovereignty for which they voted".
The deputy chairman of the European Research Group (a group of pro-hard Brexit Conservative MPs) added: "He should by all means negotiate, if necessary, until the stroke of 11 o'clock on New Year's Eve; but if the EU still refuses a deal that fully respects our hard-won independence, he should leave the table in the knowledge that he has the full support of his countrymen and women."
On Sunday morning, Conservative former minister Sir John Redwood accused the bloc of making "empty threats".
"End the talks. The EU wants to stop us being independent. They cannot stop our trade. People and businesses buy and sell under WTO rules. The EU just makes empty threats," Sir John tweeted.
"Will the Cabinet Office stop behaving as if it believes Project Fear scares? Tell us how we will use our new freedoms from EU exit to create a more prosperous UK."