"I can not see how an agreement that has been rejected three times by Parliament would not be rejected a fourth time", Mr Bone told talkRADIO's Julia Hartley-Brewer.
May is now prepared to put a tweaked Brexit deal to a vote in Parliament even if she can not guarantee it will pass. "Brings amendments into play, so it's game on again IF anything - Customs Union, EFTA, a second referendum etc - can muster a majority". "It is the same thing again, again and again".
May, who became prime minister in the chaos that followed the 2016 referendum in which Britons voted 52% to 48% to leave the European Union, is under pressure from some of her own lawmakers to set a date for her departure.
"You would have lost the loyal middle of the Conservative Party, split our party and with likely nothing to show for it", the letter said.
Asked if she would resign if the bill was defeated, May told reporters she was sure Members of Parliament (MPs) would do not forget to respect the referendum result.
May opened talks with Labour six weeks ago, acknowledging that entrenched opposition from her own side meant she needed help from across the political divide to pass her Brexit deal.
Barclay was asked by a committee of members from the upper house of parliament if this was "the last chance saloon" for May's divorce deal.
The Conservatives as a Brexit party, being very clear about their objectives are nearly certainly going to have to go into some kind of electoral arrangement with the Brexit Party, otherwise Brexit doesn't happen.
"If the House (of Commons) has not passed the Withdrawal Agreement Bill then there are growing voices in Europe, not least the French, who want to move on to other issues", he said.
An anti-Brexit protester walks outside the Houses of Parliament in London, Britain May 15, 2019.
The EU Parliament elections, which take place on May 23, could provide the ideal litmus test on where public opinion is on Brexit.
Dominic Raab, the UK's Brexit secretary, who was nominally the chief British negotiator for the deal now on the table said on Thursday: " I can not in good conscience support the terms proposed for our deal with the European Union".
Before her deal was defeated the last time, 344-286 on March 29, Ms. Mr Fox then warned MPs will be left to choose between revoking Article 50 or leaving with no deal if the Bill is voted down.
It gives legal effect to the transition period, due to run until December 2020, the rights of European Union citizens, a financial settlement with the bloc and an agreement on how to avoid a hard border in Ireland if a future trade deal with the European Union can not be concluded in time.
A Labour Party spokeswoman said Mr Corbyn set out the shadow Cabinet's concerns about the Prime Minister's ability to deliver on any compromise agreement during the talks in Parliament on Tuesday evening. "We promised people that we wouldn't be holding these elections for an institution that I voted to leave", she told the BBC.
Several Tory MPs have said they will not be campaigning ahead of the vote on 23 May and might not even vote for their local party candidate. May has turned to the Opposition Labour Party, led by veteran socialist Jeremy Corbyn, to negotiate a way out of the impasse.
Mrs May was said to have made clear to the Labour leader that she wanted to bring cross-party discussions to a conclusion and "deliver on the referendum result".