Video report by ITV News Europe Editor James Mates
Brexit negotiations remain ongoing with various different parties amid reports a deal is close and that Boris Johnson could concede ground to the EU over the Irish border.
A race against time is underway to secure a Brexit agreement by the proposed October 31 deadline and before the Brussels summit of European leaders which starts on Thursday.
The EU's chief Brexit negotiator Michel Barnier had said the two sides must agree the details by the end of Tuesday, but no formal announcement has yet been made.
Senior sources on both sides have reportedly told the Guardian that a draft treaty could be published on Wednesday morning after the UK agreed in principle there will be a customs border in the Irish Sea.
However, a spokesperson for the Prime Minister said "talks remain constructive but there is more work still to do".
Downing Street sources were also downplaying the chances of a breakthrough being imminent and an EU official stressed "talks are ongoing".
ITV News Political Editor Robert Peston analyses the chances of a Brexit deal being reached
Meanwhile, Irish Taoiseach Leo Varadkar said it remained uncertain whether a deal would be ready in time for the Brussels summit on Thursday and Friday of this week.
"The initial indications (from the EU) are that we are making progress, negotiations are moving in the right direction," he told reporters.
"But whether we will be able to conclude a revised Withdrawal Agreement, which is an international treaty, in time for the summit, that's as of now unclear."
Mr Varadkar also revealed that the PM told him during their meeting last week he was "confident" he would be able to do what Theresa May thrice failed to do by getting a deal through the House of Commons.
Suggestions a deal was closing in came as Downing Street declined to recognise a midnight deadline apparently set by EU chief negotiator Michel Barnier.
Mr Barnier warned Mr Johnson that "it is high time to turn good intentions into legal text".
But the PM's spokesperson said: "We are working hard. The Prime Minister is aware of the time constraints that we are under."
Various members of the European Research Group (ERG) of Tory Brexiteers attended a meeting inside Downing Street on Tuesday afternoon.
The group's head, Steve Baker, told ITV News Political Correspondent Paul Brand he was optimistic a "tolerable" Brexit deal could be agreed and that talks had been "constructive".
However, less-positive was fellow Brexiteer Mark Francois who said the meeting was "interesting" and added "there'll be further chats to have", while former Tory leader Iain Duncan Smith and MP Sir Bill Cash also emerged from Downing Street.
As well as the ERG, members of Northern Ireland's Democratic Unionist Party (DUP) also held talks in Downing Street, but left via a backdoor to avoid media questions.
Following the meeting which lasted for an hour-and-a-half, the party put out a statement saying they "respect the fact negotiations are ongoing, therefore cannot give a detailed commentary, but it would be fair to indicate gaps remain and further work is required".
Mr Johnson is set to update his Cabinet on Brexit on Wednesday afternoon in order to give them the most up to date information on talks, and later address the 1922 Committee of Tory backbenchers.
There have been some suggestions that this could indicate a deal or draft agreement could be unveiled.
Downing Street officials are understood to have been meeting with various parliamentary factions in recent days as negotiators hammer out a deal.
Mr Barnier struck a positive note after meeting Brexit Secretary Stephen Barclay at the General Affairs Council on Tuesday morning.
He debriefed EU27 ministers in Luxembourg before tweeting: "Talks are difficult but I believe an agreement is still possible."
The latest comments comments came as negotiators stepped up efforts to work out a way to break the deadlock over the Irish backstop, the contingency measure to prevent a hard border on the island.
Irish broadcaster RTE had reported that two sources confirmed that British negotiators will bring forward an updated plan on Tuesday to deal with the issue of customs and the Irish border.
Ireland's deputy premier Simon Coveney stressed "significant progress" would need to be made on Tuesday "if there is to be a deal that Michel Barnier can report on tomorrow to EU capitals in advance of the leaders' summit".
Meanwhile, Downing Street said Mr Johnson told French President Emmanuel Macron in a phone call that UK officials would "continue to work hard" on securing a Brexit deal.
The two-day EU summit is crucial because the PM must get a new deal approved by MPs by Saturday if he is to avoid a clash over asking for a Brexit delay.
The Benn Act passed by MPs opposed to a no-deal, including Tory rebels, says he must ask for an extension to Article 50 if MPs do not back a deal by then.
There are fears that a loophole could be used to avoid this, with the PM repeatedly ruling out making the extension request under his "do or die" pledge to get Brexit done by the Halloween deadline.
Leader of the Commons Jacob Rees-Mogg has suggested the Government could use European law to achieve no deal, while Labour has threatened court action to force the PM to obey the legislation.
Regardless of the outcome in Brussels, a showdown is anticipated in an emergency sitting of Parliament on Saturday, the first in 37 years, if the Government requests the unusual move and it is backed by MPs.
They will be able to back or reject any deal presented to them, or there will be discussions on what to do next in the Brexit saga.