Brexit secretary Mr Barclay met Mr Barnier for dinner in Brussels on Monday night. Both sides claimed “constructive” talks were held, however the government is no closer to breaking the impasse.
Mr Barnier told reporters afterwards: “It’s clear from our side we are not going to reopen the Withdrawal Agreement, but we will continue our discussions in the coming days.”
Mr Barnier said Theresa May should endorse a permanent customs union - as proposed by Labour - to break the impasse over their looming divorce.
"I found Corbyn's letter interesting in tone and in content," Barnier said earlier in the day in Luxembourg. "Something has to give on the British side."
Barnier said he understood May did not want to delay Brexit to win more negotiating time and reiterated the controversial Irish border fix - or "backstop" - in Britain's withdrawal agreement was an insurance policy the EU hoped never to use.
"This time that remains is extremely short," Barnier told the joint news conference with Luxembourg Prime Minister Xavier Bettel. "The withdrawal agreement which we agreed with Theresa May's government ... remains the best way to ensure an orderly withdrawal of the UK."
"We are determined to organise an orderly withdrawal. This orderly withdrawal is the prerequisite that we need to build a ... future relationship," said Mr Barnier.
Michel Barnier after meeting UK ministers tonight: "We will not open the Withdrawal Agreement but we will continue our discussions in the coming days." pic.twitter.com/ZA9zOglq25— Mark Johnston (@MarkJohnstonLD)February 11, 2019
Back in London, meanwhile, Brexiteer Cabinet minister Andrea Leadsom insisted there was “no chance” of Prime Minister Theresa May accepting Labour's vision for Brexit.
It followed speculation Mrs May could soften her stance on customs union membership – a key aspect of Jeremy Corbyn’s demands.
But she told the Press Association: "I think she's making quite clear that what Corbyn is demanding is actually not as good as what the Prime Minister's deal is offering.
"So he wants a customs union and he is unclear as to whether that means he also wants an independent trade policy.
“He's unclear as to whether he also wants to stop free movement, and of course the EU's view would be well if you're in the customs union then you have free movement and you abide by the common external tariff.
"I think there's no doubt that what the Prime Minister is offering is better than what Corbyn is demanding, which simply begs the question: if they like it, why don't they vote for it?"
She added: “The Prime Minister has been absolutely clear we're leaving the EU, we're leaving the customs union, we're leaving the single market.
"We're taking back control, we won't be paying money over, free movement will end, and we will have our own independent free trade policy. So I definitely don't see the Prime Minister agreeing to Corbyn's world view."
Mr Corbyn faced fresh scrunity over his Brexit stance on Monday, after more archive footage emerged of him slating the EU. It prompted leading Brexiteer Nigel Farage to remark: "I feel like a Europhile compared to Corbyn. Wow!"
On the government's side, meanwhile, Chris Grayling faced fresh calls to resign in the Commons as he defended the government's decision to issue a £13.8million ferry contract to a company - Seaborne - which had no ships.
Additional reporting by Press Association.