The report further said that senior party sources said they believe two-thirds of Labour MPs, including several shadow cabinet ministers and many more frontbenchers, would refuse to back a deal without a people's vote attached.
"To the Leader of the Opposition I say this: Let's listen to what the voters said in the local elections and put our differences aside for a moment".
Former Conservative Party leader Iain Duncan Smith - who was himself ousted by his own MPs in 2003 - argued it was time to change the leadership, saying backbench MPs needed to "decide that either the prime minister sets the immediate date for departure or, I'm afraid, (we) must do it for her".
May's Conservatives lost more than a thousand seats on English local councils that were up for re-election, and Labour, which would typically aim to gain hundreds of seats in a mid-term vote, instead lost 81.
The Observer says both party leaders were facing mounting revolts from within their ranks, with opposition MPs and Tory Brexiteers warning any deal would face inevitable defeat in Parliament and cause more acrimony.
The broadsheet said May would set out plans for a temporary customs arrangement with the European Union that would last until the next general election, which must be held by May 2022.
"We've got to be a broad party. I actually think she's jeopardised the negotiations for her own personal protection".
"The very worst thing we could do at this time is a Westminster stitch-up, whether over the PM's deal or another deal".
Sir Graham Brady, the chairman of the 1922 committee of Tory backbenchers, told the Telegraph that staying in a customs union could lead to a "catastrophic split" in the Conservative Party.
Complicating the picture, the main beneficiaries of the swing against the two major United Kingdom parties were the pro-EU Liberal Democrats, who campaigned on a demand for a new referendum, aiming to reverse Brexit.
But shadow chancellor John McDonnell accused the Prime Minister of jeopardising the talks, claiming she had "blown the confidentiality" of the discussions. "I can't think that any other party in the United Kingdom has raised money like that". "A lot of this rests, to be honest, on one man, on whether Jeremy Corbyn really wants to deliver a Brexit deal", Stewart told Sky News.
"But I don't think we should give false hope on this, it's going to be very hard to find a negotiated settlement".
Ruth Davidson, leader of the Scottish Conservatives, on Saturday pledged to resist any new referendum on independence from the United Kingdom just as support for secession rises.