Housing minister James Brokenshire said on Sunday May would commit to giving parliament another debate on Brexit with the chance to vote on alternative options, if a deal had not yet been agreed and voted upon by then.
In her letter, Mrs May said she wanted discussions between Tory and Labour teams to start considering "alternative arrangements" to the backstop contained within the withdrawal agreement.
The intervention came as the PM (pictured left returning to No10 today) moved to quell Tory fears that she is about to cave into Jeremy Corbyn's (pictured right) demand for a permanent customs union with the EU.
The prime minister's formal response to Jeremy Corbyn's proposal, in a letter to the Labour leader, stressed her objections to keeping the United Kingdom in some form of customs union, saying this would prevent the United Kingdom making its own trade deals.
But Mr Johnson insisted that a separate codicil setting out amendments to the backstop plan would not be "good enough", indicating that he is holding out for changes to the text of the agreement itself.
A top European Union official warned Monday (4 February) that the prospect of Britain crashing out of the bloc next month looks more likely than ever, as Prime Minister Theresa May sought to reassure foreign investors anxious about a chaotic Brexit.
That might mean that she's back before the end of this month with something concrete to vote on.
Last week, Labour leader Jeremy Corbyn set out the conditions under which he would instruct his party to support an exit deal in parliament.
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The letter concludes with Mrs May saying she looked forward to the two parties meeting "as soon as possible".
"This new agreement lays the foundations for the two parties to continue their sound economic and trade relations once the United Kingdom has left the European Union", the Swiss government said in a statement (here in English). He has also faced pressure from some of his MPs to push for another public vote on Brexit.
"The point of the exercise is to say there is a majority for a close economic relationship if you are prepared to try to find it and I said we should test that by having a vote on the customs union", Sir Keir said.
"I don't see that in the letter", he said.
"If you are going to stay in the customs union and the single market and you are going to have people around the table in Brussels, there's a name for that".
She said the government needed to "get serious" about policy on a customs union and guarantee to "legislate for the protections around workers' rights".
But May's office published her reply to Corbyn late on Sunday (10 February), showing little appetite for a U-turn which would risk splitting her fractious party by ruling out the scope for Britain to strike its own trade deals around the world.
Mr Reidy said: "It's really important that a consensus is built across the House of Commons but a credible consensus that also gets the assent and agreement of the EU27".