The move comes after Mrs May responded to Mr Corbyn's letter last week setting out Labour's terms for backing a deal, with an offer of further talks.
The International Trade Secretary Liam Fox dismissed Labour's plans as unworkable, while Boris Johnson warned that such a deal would have a disastrous effect on the Conservative party.
Rather than insisting it is removed altogether, however, former foreign secretary and two-time London mayor Johnson said Monday he could support the Prime Minister if she was able to get the European Union to agree to a time limit or other exit mechanism for the backstop.
But senior Tories favouring a hard Brexit were angry she had refused to rule it out completely and had left the door open for more talks.
She said the government needed to "get serious" about policy on a customs union and guarantee to "legislate for the protections around workers' rights".
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Asked if changes to the backstop proposals could come in a separate codicil to the Withdrawal Agreement, Mr Johnson said: "I don't think that would be good enough".
It comes after Mr Corbyn last week set out five Brexit demands to the Prime Minister, among which included the United Kingdom joining a permanent customs union with the European Union.
Mrs Leadsom said there was "no chance" Mrs May would adopt Mr Corbyn's "view of the world", adding: 'The Prime Minister has been absolutely clear we're leaving the EU, we're leaving the customs union, we're leaving the single market'.
But she said it also recognises the development of the UK's independent trade policy.
She added: "The fundamental negotiating challenge here is the EU's position that completely frictionless trade is only possible if the United Kingdom stays in the single market".
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"I think that gives that sense of timetable, clarity and goal on what we are doing with the European Union - taking that work forward and our determination to get a deal - but equally knowing that role that Parliament very firmly has", he said on BBC's Andrew Marr Show.
Mr Barclay will later travel to Brussels for talks over dinner with the EU's chief negotiator Michel Barnier.
And instead of a "meaningful" vote, ministers could be given another series of non-binding votes on possible Brexit alternatives by February 27 with the final vote on whether to approve or reject the deal delayed until the following month.
She has said she wants to bring a revised deal back to parliament for a vote "as soon as possible" but has not yet set a date for doing so.
"That gives that sense of timetable, clarity, and objective on what we're doing with the European Union - taking that work forward and our determination to get a deal - but equally knowing that role that parliament very firmly has", Brokenshire told the BBC.
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Red Bull's second team will be the first to reveal next week, and the big news is Daniil Kvyat's Lazarus act. I will always try to get the best out of myself and hopefully that is enough".
May is already due to update parliament on her progress towards a deal on Wednesday and then on Thursday to give parliament a chance to express their opinion.