Labour MP Yvette Cooper who, along with Conservative Nicky Morgan, is behind the amendment, said Parliament must act now to rule out a no deal in the event of Mrs May's agreement being voted down next week and MPs being unable to agree any other course of action before the UK's exit in March.
British MPs will resume their parliamentary debate on Brexit on Wednesday, with a vote on the deal expected on Monday or Tuesday next week.
But Mr Corbyn hailed the development as an "important step" to prevent a no-deal Brexit.
The chances of the United Kingdom crashing out have risen dramatically as the PM faces the prospect of disastrous defeat in a vote next week on the deal she has thrashed out with the EU.
Reports this weekend suggest that Downing Street are planning to put the deal before parliament up to 30 times in an attempt to bludgeon MPs into backing it in order to prevent a no-deal Brexit.
A deal setting out the terms of the United Kingdom's divorce from the European Union has been agreed to between the prime minister and the European Union, but it must pass a vote in Britain's Parliament before it is formally adopted.
He said May has had further conversations with European Union leaders over the holiday period, adding that the government will set out the assurances it had received before the debate starts on Wednesday.
Britain's divorce deal with Brussels is the only deal on the table and can not be renegotiated, an EU Commission spokesperson said Monday.
The issue of personal safety is seen as vital in view of the assassination of Labour Party lawmaker Jo Cox during the Brexit referendum held in June, 2016.
She said: "No-deal is a bad deal and it would be a gross dereliction of the responsibility of members of this House to inflict a no-deal situation on our constituents".
"The backstop remains the poison which makes any vote for the Withdrawal Agreement so toxic", DUP Deputy Leader Nigel Dodds said in a statement on Sunday.
There was little sign that opposition to Mrs May's deal among Tory Brexiteers had softened over the Christmas break, with several who attended a drinks reception at Number 10 on Monday evening saying they would vote against the plan.
The vote had originally been scheduled for December 10, but May postponed it, admitting in the House of Commons that it would have been "rejected by a significant margin".
Johnson said, "we must hope that Theresa May does remove the backstop from the withdrawal agreement, in such a way as to give real legal protection to the UK".
Twenty Tory rebels MPs defied the whips to back the change - including former ministers Kenneth Clarke, Sir Michael Fallon, Justine Greening, Dominic Grieve and Sam Gyimah.
"If we as a new, incoming Labour government were to go to Europe without those red lines we know that we could get a different, better deal and that's what we want to try and achieve".
"The closer we get to 29 March without a deal, the more assets will be transferred and headcount hired locally or relocated", Omar Ali, UK financial services leader at EY, said.
May has repeatedly said that the alternatives to her Brexit deal are "no deal" or "no Brexit".
"I'm anxious we could come to the crunch and parliament will not have the powers to stop [no deal] happening", she said.
But Mr Fox, who backs Mrs May's deal, said it would be "irresponsible to tie the government's hands" at this stage by ruling out any options.