The Conservative MP met with the EU's chief negotiator, Michel Barnier, in the hope of making a breakthrough in talks and avoid a no-deal.
Mr Barclay said: "I think there is a shared desire, reflected in the meeting today, to secure a deal because we both recognise that a deal is in the interest of both sides.
"No one wants to see a no deal. It would be disruptive to both sides and that's why there's a shared, sense of purpose to get a deal over the line."
Mr Barclay pointed out that the scheduled meeting with the EU overran its allotted time frame, which he suggested that progress was being made.
"A very clear message has been given, both by president Juncker and the prime minister, who want to reach a deal.
"The meeting actually overran which I think signals that we were getting into the detail."
He confirmed a further round of talks will take place with the EU next week.
On Thursday, Mr Juncker said his meeting with Boris Johnson in Luxembourg on Monday was “rather positive” as he assured he was “doing everything to have a deal” to prevent a “catastrophic” no-deal Brexit.
Speaking to Sky News, Mr Juncker said “we can have a deal”, but was unable to put the chances at more than 50/50 when pressed.
“If the objectives are met – all of them – then we don’t need the backstop,” he added.
The pound reached a two-month high following Mr Juncker’s comments.
Mr Barclay’s department said he will meet with Mr Barnier to “take stock” following discussions between the PM’s Europe adviser David Frost and Taskforce 50 – the EU unit dealing with the UK’s departure.
The EU’s original proposal, which was objected to by the DUP, would see Northern Ireland remaining in the EU’s single market and customs union, but give Great Britain the freedom to strike trade deals.
Previously documents had been shown to Brussels officials, but then taken back at the end of meetings out of fears they would be leaked.
But a “non-paper” is not a formal Government position and falls far short of what has been demanded by Brussels.
The PM was under pressure from Finnish prime minister Antti Rinne to formally outline his plans to the EU by the end of September.
Mr Rinne told reporters after meeting French president Emmanuel Macron that they “agreed that it is now time for Boris Johnson to produce his own proposals in writing – if they exist”.
The Government insisted Mr Johnson will not be bound by an “artificial deadline” to produce formal written proposals to resolve the Brexit deadlock.
Mr Johnson said progress was being made, but the extent of it should not be exaggerated.
“You heard Jean-Claude Juncker yesterday say that he doesn’t have any emotional attachment to the backstop,” Mr Johnson said.
“Now that is progress – they weren’t saying that a month ago.”