The UK has been urged to submit fresh proposals within the next 48 hours to break the Brexit impasse.
EU officials said they would work non-stop over the weekend if “acceptable” ideas were received by Friday to break the deadlock over the Irish backstop.
The UK has said “reasonable” proposals to satisfy MPs’ concerns about being tied to EU rules had already been made.
Chancellor Philip Hammond has warned Brexiteers to vote for the PM’s deal or face a delay to Brexit.
The PM is seeking legally-enforceable changes to the backstop – an insurance policy designed to prevent physical checks on the border between Northern Ireland and the Republic of Ireland, but there have been few visible signs of progress.
MPs are due to vote for a second time on the Brexit deal next week. If they reject the deal again, they will get to choose between leaving without a deal or deferring the UK’s exit from the EU beyond the scheduled date of 29 March.
Speaking to BBC Radio 4’s Today programme, Mr Hammond refused to be drawn on how he would vote if Mrs May’s deal is defeated.
“If the prime minister’s deal does not get approved on Tuesday then it is likely that the House of Commons will vote to extend the Article 50 procedure, to not leave the European Union without a deal, and where we go thereafter is highly uncertain,” he told BBC Radio 4’s Today programme.
“For those people who are passionate about ensuring that we leave the European Union on time it surely must be something that they need to think very, very carefully about now because they run risk of us moving away from their preferred course of action if we don’t get this deal through.”
Mrs May is pinning her hopes on getting changes to the backstop that will prevent the UK from being tied to EU customs rules if no permanent trade deal is agreed after Brexit.
Critics say that – if the backstop were used – it would keep the UK tied to the EU indefinitely.
Negotiations between British ministers and the EU officials over the past 24 hours have been described as “difficult”, with the EU insisting there has been no breakthrough.
Diplomats from the 28 member states were told on Wednesday that Mrs May could meet European Commission President Jean-Claude Juncker on Monday if progress was made.
But the BBC’s Europe reporter Adam Fleming said talk of a 48-hour deadline for new proposals and a weekend of negotiations was “a notional timetable” and that more flexibility could be possible.
Attorney General Geoffrey Cox, who is leading the UK team, has conceded that negotiations are at a sensitive point and the exchanges have been “robust”.
Mr Cox, who will take questions from MPs on Thursday, has played down reports he has abandoned hopes of getting the EU to agree to a firm end date to the backstop or some kind of exit mechanism – key demands for many Tory Brexiteers.
