Writing her response to his letter of last Wednesday, Mrs May told the Labour leader: "It is good to see that we agree that the United Kingdom should leave the European Union with a deal and that the urgent task at hand is to find a deal that honours our commitments to the people of Northern Ireland, can command support in Parliament and can be negotiated with the EU - not to seek an election or second referendum".
"So he wants a customs union and he is unclear as to whether that means he also wants an independent trade policy".
In her response in the letter, the Prime Minister said she wanted the Tory and Labour teams to consider "alternative arrangements" to the Irish backstop.
May has been seeking changes to the deal with Brussels since it was rejected by a record majority in parliament on January 15.
Jacob Rees-Mogg, chair of the European Research Group of eurosceptic Tory MPs, told The Daily Telegraph: "As the 2017 Conservative manifesto promised to leave the customs union, it would be more consistent for the prime minister simply to rule one out".
Treasury Chief Secretary Liz Truss refused to rule out resigning if Mrs May backed a customs union.
But it would prevent Britain from making its own trade deals after Brexit.
Cabinet will meet this morning, with Downing Street hoping it has seen off any immediate risk of resignations over the looming prospect of a no-deal Brexit by promising a further round of Brexit votes at the end of February, if Mrs May still hasn't achieved the renegotiated deal she is calling for from Brussels.
At the start of another busy week in the Brexit process, Mrs May spokesman was quizzed about what happens next at a daily Westminster briefing with journalists.
The government announced on Sunday that parliament might not be given the chance to vote on its Brexit deal until March.
British lawmakers overwhelmingly rejected the deal, agreed in November, in a vote last month, leaving Britain on course for an abrupt Brexit with no transition period to minimise expected economic disruption.
Liam Fox, the International Trade Secretary, described Labour's plan as a "dangerous delusion", while Boris Johnson, the former foreign secretary, accused Mr Corbyn of trying to trap Mrs May into a "toxic" Brexit.
The Prime Minister will ask Parliament not to try and force an extension to Article 50 when the Commons is presented with another "neutral motion" on Brexit later this week, rather than a substantive vote to approve the Brexit deal.
He criticised Labour's customs union proposals, saying they were "not workable".
Brexit minister Stephen Barclay will meet European Union negotiator Michel Barnier on Monday to discuss changes to the part of the exit deal relating to the "backstop", an insurance policy against the return of a hard border between European Union member Ireland and British-ruled Northern Ireland.
Mrs May is trying to win concession from Brussels on the controversial backstop, the insurance policy created to avoid a hard border returning to the island of Ireland.
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