Cabinet is meeting this week, with suggestions that preparations will be stepped up even further for no-deal, and pressure building on Theresa May to give a firm date for departing as other senior conservatives including Boris Johnson and Esther McVey start to discuss their own plans to run for leader.
International Development Secretary Rory Stewart said this weekend that the government and Labour were "half an inch apart" on key issues.
The Scottish National Party and ChangeUK said they would not back her. Nigel Dodds, the deputy leader of Northern Ireland Democratic Unionist Party, which props up May's Conservatives, said the proposal had "fundamental flaws". "But this speech from the PM means there is no way I will support the Withdrawal Agreement Bill".
"The EU has said very clearly that the Withdrawal Agreement has been negotiated over two-and-a-half years, it was agreed with the British government and the British cabinet and it's not up for renegotiation, even if there is a new British prime minister", he said.
"The Labour and Conservative positions are about half an inch apart", he told BBC1's The Andrew Marr Show.
"Despite Mr Corbyn ending the talks last week, Mrs May has come back with something that we think is going to include some fresh details".
The DUP, on whose support May depends to command a majority in parliament, says efforts to avoid a "hard border" between Northern Ireland and the Republic of Ireland - keeping Northern Ireland largely aligned with the European Union until a new trade deal can be agreed, also known as the "backstop" position - undermines the integrity of the United Kingdom, creating a trade border between Northern Ireland and Great Britain.
Speaking at the headquarters of PricewaterhouseCoopers, May appealed to lawmakers to get behind her deal, offering the prospect of a possible second referendum on the agreement and closer trading arrangements with the European Union as incentives.
There would then be a leadership election, in which some analysts fear a hardline candidate prepared to leave the European Union without a deal could win over Conservative MPs and members.
Writing in the Sunday Times, May said she will make "a bold new offer" to members of Parliament.
May was scheduled to outline her plan in a speech outside Downing Street later on Tuesday, her office said, after lawmakers in parliament's elected main house, the Commons, rejected her deal three times.
Before then, the government is considering holding a series of "indicative votes" to see what path, if any, lawmakers might be able to agree.
May said Friday that British lawmakers "will be faced with a stark choice: that is to vote to ... deliver Brexit, or to shy away again from delivering Brexit with all the uncertainty that that would leave".
Theresa May will set out a "bold new offer" on her Brexit deal this afternoon, aimed at winning the support of Labour MPs.
"So the new Brexit deal goes further", she said.
Another defeat would nearly certainly see a ratcheting up of demands for her to go immediately, amid intense frustration at her failure to deliver on the 2016 referendum result.
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