Either way, the chances are increasing that Parliament will get a greater say over Brexit.
The BBC's political editor Laura Kuenssberg said she had been told by a government source that no actual concessions had been agreed, and the only agreement was to keep talking.
Former Tory cabinet minister Dominic Grieve, who tried to broker a compromise between MPs and ministers, has warned a no-deal scenario could lead to a "national crisis".
The debate, which lasted for almost three hours, was split down the usual non-partisan lines that have emerged as a result of Brexit, with the likes of Labour's Kate Hoey and John Mann saying they would back the Conservative government, while Tories including Ken Clarke and Anna Soubry spoke in favour of Grieve.
The government was putting a combative spin on the concessions Tuesday evening: "The Brexit Secretary has set out three tests that any new amendment has to meet - not undermining the negotiations, not changing the constitutional role of Parliament and Government in negotiating global treaties, and respecting the referendum result", a spokesperson for the Brexit department said in a statement.
Parliament will vote Tuesday on a key piece of legislation, the E.U. Withdrawal bill, that would transfer European Union laws now on British books into British law after Brexit.
GBP/USD has rallied to test the bear's commitments at the top of the descending channel at resistance on the back of the government winning the third vote as MPs vote on Lords amendments to the European Union withdrawal bill. "There is goodwill to secure the protection we are seeking in the event of no deal".
The current "meaningful vote" offered by the Government allows MPs to accept or reject the Brexit deal, but the proposed change would see MPs take control of the negotiation aims.
Hours before the debate began, a justice minister resigned in protest at what he called its "wish to limit" the role of parliament in shaping Brexit.
I am concerned obviously because amendment 19 from the House of Lords was defeated yesterday and any derivative of it is an unashamed wrecking motion to Brexit. "So we are partway through discussions and there is more to come", she said.
"I trust the prime minister".
Solicitor General Robert Buckland said the government would remain open-minded, but the meeting may not result in new proposals in the coming days.
"A vote between bad and worse is not a meaningful vote". A paper laying out the U.K. government position, due to be published this month, has been delayed because the Cabinet can not agree on a united stance.
He rejected suggestions ministers were already backtracking on promises made to MPs, insisting he did not want to raise any "false expectations".
Earlier, the prime minister had been hit by the resignation of justice minister Phillip Lee, who came out in support of a second referendum on the UK's final divorce deal and criticised Mrs May's Brexit policy.
And in a final vote, MPs voted by 326 votes to 301 to disagree with Lords amendment 52 linked to what European Union laws the United Kingdom will keep after Brexit.
Labour's Shadow Brexit Secretary Keir Starmer said that the only reason May's government had decided on a climb down was because 'they thought they were going to lose the vote'.