Mr Lidington confirmed the final issues still to be resolved included the so-called Northern Ireland "backstop", meant to ensure there is no hard border in the event of a no-deal Brexit. But as domestic opposition builds, momentum seems to be fading for getting a deal done quickly.
On Monday night, some cabinet ministers met for drinks in International Trade Secretary Liam Fox's office to discuss Brexit, including no-deal plans and the sticking point of the Irish "backstop".
The college of commissioners will meet in Strasbourg where they will receive an update on Brexit negotiations from Mr Barnier.
"In these final stages of the negotiations, ministers showed again today that we are determined to keep the unity of the EU27", said Austria's Arts and Culture Minister Gernot Blumel. It is unclear who might replace her.
Theresa May will push the button on no deal Brexit planning if there is no breakthrough in talks this week.
In an op-ed in The Times, he wrote: "All of us want to be able to look our constituents in the eye and say that leaving the European Union will work out fine, but more and more of us, including a number in the cabinet, are now convinced we are heading for a poor outcome for the country". Northern Irish lawmakers are concerned the region will end up being treated differently to Britain.
Boris Johnson's younger brother Jo sensationally quit the Cabinet last week and laid into the PM's plans - going as far as calling for a second referendum.
"I will not compromise on what people voted for in the referendum", she said. Delay after staged managed delay.
He added that May was forcing Brexit two choices on the country - her deal or no deal - and that "to present the nation with a choice between two deeply unattractive outcomes, vassalage and chaos, is a failure of British statecraft on a scale unseen since the Suez crisis".
The source said that a deal would need to be secured by the end of play on Wednesday, if a summit was to be agreed this month.
On Monday Mayor of London Sadiq Khan, an influential figure in the Labour Party, urged his party in the House of Commons to reject May's deal.
Irish Foreign Minister Simon Coveney saw no substantive progress after meeting Barnier.
EU's chief Brexit negotiator Michel Barnier updated the top diplomats on the complicated negotiations with the United Kingdom government, which several ministers, including Belgium's Didier Reynders, said were nearing an end, according to Efe news.
The Brexit process, and the "toxic" alliance between the Conservatives and DUP, had contributed to progress made in the Good Friday Agreement being "thrown under the bus", Ms O'Neill said.
Pro-Brexit Commons Leader Andrea Leadsom added to Tory tensions by insisting the United Kingdom could not be "trapped" in a backstop agreement without the ability to leave at a time of its choosing. "We assess the probability of a deal to trigger the status quo transition period as at least 70 percent".
"It would cost us money but it would finally dispel the "crash out" Project Fear nightmare scenarios", he wrote in the Mail on Sunday.
Even with such measures in place, the British government says leaving the European Union without a deal could cause major economic disruption, with gridlock at ports and disruption to supplies of foods, goods and medicines.
European Union officials have repeatedly insisted that they will not give in to demands over the Irish backstop - an insurance measure created to avoid a hard border in Ireland - which they see as integral to any divorce deal.