If her deal is rejected for a third time and is not approved by Wednesday, the government will ask for a longer extension.
MPs in the House of Commons voted 412 to 202 on Thursday in favour of a government motion proposing a "one-off extension. for the objective of passing the necessary European Union exit legislation" until June 30, provided May's widely maligned deal wins parliament's approval by Wednesday next week.
Parliament's votes this week won't end Britain's Brexit crisis.
Meanwhile, British Prime Minister Theresa May has piled renewed pressure on reluctant MPs to back her European Union divorce deal at the third time of asking.
MPs have proposed changes to that including one, amendment H, which says a delay should be used to hold a second referendum.
"I am very, very suspicious and concerned about the time scale", Conservative lawmaker Andrew Bridgen said.
And by the far narrower margin of 314-312, they voted down a cross-party bid for parliament to seize control of the Brexit process.
The Defence Secretary was one of seven Cabinet Ministers to vote against the Government in opposition to plans to delay Brexit beyond March 29.
May's plan is to hold a third vote on her agreement with Brussels by Wednesday. She is seeking to win over opponents in her own party and its Northern Irish political ally, the Democratic Unionist Party.
That gives her three days next week to overturn the 149-vote defeat she suffered on March 12.
"It's still really hard to see how the numbers stack up for Theresa May, but she's giving it one more go", he said. Britain will need to show how it could use more time to find a way forward, when it has so little to show for two-plus years of political infighting. May hopes that this can convince enough MPs to start supporting her deal.
The European Commission said a delay required the unanimous agreement of all 27 member states and it was up to the the European Council to consider the request at its summit on 21 March.
And, on the right, conservative bloc leader Manfred Weber chided Britain's Labour for opposing May's deal. The main pro-Brexit faction in May's party, the European Research Group, said it did not recommend voting for her deal.
The longer extension will mean that Britain must also participate in the upcoming European Parliament elections.
Belgian Prime Minister Charles Michel said in Brussels said he wasn't sure more time was the answer.
Anxious businesses are pleading for action and US President Donald Trump waded in to pronounce himself "surprised to see how badly it has all gone". It depends in my view on what conditions the European Union places on that extension, because of course the British government now is tasked by parliament to request an extension, and they will undoubtedly provide a time period on that, so they will be asking for an extension for a particular period of time. With the approaching deadline intensifying fears that Britain could leave the bloc without a deal - a move that economists say could spark economic turmoil - Parliament voted to rule out the possibility of a no-deal Brexit.
European Council President Donald Tusk said on Twitter that he would appeal to the 27 EU leaders "to be open to a long extension if the United Kingdom finds it necessary to rethink its Brexit strategy and build consensus around it", while French President Emmanuel Macron said that the extension could only be "a technical delay to allow more time to put their departure in place". "Something has to give".