Late-night UK-EU talks in Brussels ended with "optimism on both sides", government sources say, although no final Brexit deal has been agreed.
The Northern Ireland border remains the key issue to be resolved.
Prime Minister Theresa May is updating her cabinet on the latest developments at a meeting in Downing Street.
Addressing the Lord Mayor's Banquet on Monday evening, the prime minister said the talks were "immensely difficult" but were now in the "endgame".
No 10 says Mrs May will not have a new Brexit plan to put before the cabinet and will instead update them on the state of the negotiations, especially issues over how best to guarantee that there is no return to a visible border on the island of Ireland.
Cabinet Office minister David Lidington, who deputises for Mrs May at Prime Minister's Questions, told BBC Radio 4's Today "we are almost within touching distance" but warned that "it can't be a deal at any price".
Asked to comment on reports that the UK would have to give the go-ahead to major spending on no-deal preparations if there was no agreement by Wednesday, Mr Lidington said "there does come a time at which contingency planning does have to be stepped up".
BBC assistant political editor Norman Smith said it was thought the only "substantive" area of disagreement left was over the mechanism for the UK to exit the proposed "backstop" customs arrangement - referring to the fallback plans to guarantee there will be no new visible border checks.
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 Irish "backstop".
Unease over the backstop centres on how it will work.
The prime minister is under pressure from both Brexiteers and pro-EU MPs as she tries to seal a deal with the EU on the terms for the UK's exit on 29 March 2019.