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."
The chancellor's explicit warning to Brexiteers
By Laura Kuenssberg, BBC political editor
What we heard from the chancellor this morning was that he was clear about the uncertainties ahead - and rather unclear (cagey, in fact) about how he might vote when it came to decision-time about a no-deal.
There was an explicit warning to Brexiteers: vote for the prime minister's deal because otherwise, it's delay and a soft Brexit.
As one minister expressed to me yesterday, they believe the vote does have a chance of getting through because Brexiteers will realise - just in time - that it's either the PM's deal next week, or what this minister described as "soft, softer, then meltdown".
But across government, the mood is not optimistic about what's going to happen next week and most ministers are expecting a defeat.
French Europe minister Nathalie Loiseau reiterated the EU's position that the withdrawal agreement cannot be reopened and said the deal was the "best possible solution" with the controversial Irish backstop a "last resort solution".
She said: "We don't like the backstop, we don't want to have to implement it, and if we have to, we don't want to stay in the backstop.
"We all agree that it should be temporary."
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.
Arbitration plan founders
By BBC's Brussels reporter Adam Fleming
The latest talks aimed at securing legal guarantees about the Irish backstop foundered over a British proposal for the role of the independent arbitration panel which will be set up under the Brexit deal.
It will be made up of judges and lawyers, and will handle disputes between the UK and the EU about the withdrawal agreement.
The British suggested it have a role in deciding whether the backstop should come to an end - if it's ever needed.
But the EU felt that went beyond the panel's remit, which is to ensure each side sticks to the rules - not to make big decisions like the future of the Irish border.
Hence the request for the UK to think again. And quickly.
Jeremy Corbyn has met Conservative MPs to discuss possible alternatives to the PM's deal.
The Labour leader held talks with ex-Tory minister Nick Boles and Sir Oliver Letwin, who favour a closer, Norway-style relationship with the EU.
He said he had discussed the so-called "Common Market 2.0 option" - which would see the UK remain in the EU's single market by staying part of the European Economic Area - but would not commit to backing it at this stage.
The government has suffered the first of what are expected to be a number of defeats in the Lords on a key piece of post-Brexit legislation.
Trade Bill defeats
Peers voted to amend the Trade Bill to call on the government to join a new customs union with the EU after Brexit.
The result means MPs now will get a vote on whether to stay in the existing customs union when the legislation returns to the Commons.
Ministers also lost a vote obliging them to get Parliament's approval for its negotiating strategy ahead of the next phase of talks on future relations with the EU.
Meanwhile, Mr Corbyn said he had agreed to meet Conservative MPs because he was adamantly opposed to a no-deal exit and he wanted to hear "what their ideas and options are".
While Labour wanted an agreement encompassing a customs union, unhindered access to EU markets and legal protection of workers rights, he said that "what exact form that takes is subject to negotiation".
Mr Boles said the goal was to reach a cross-party compromise to ensure the UK left the EU but in a manner which protected its economic interests.
Oliver Letwin and I were happy to join @LucyMPowell and @SKinnock in this meeting with Jeremy Corbyn to discuss our Common Market 2.0 proposal. For months now we have been meeting with senior MPs from all parties in search of a cross party Brexit compromise. https://t.co/dNKpLbEQXR— Nick Boles MP (@NickBoles) March 6, 2019