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.
And Mrs May wrote in the Mail on Sunday: "To the leader of the opposition, I say this: let's listen to what the voters said in the elections and put our differences aside for a moment".
"There is a deal that could be done in the next few days. and I really hope we can get to that point", she said, describing the results as "a kick up the backside".
The Conservatives are desperate to move forward after losing hundreds of positions in last week's local elections.
The Sunday Times reported that the Conservatives would offer new concessions to Labour when talks restart on Tuesday, including a temporary customs union with the EU, which would last until a national election due in June 2022.
Asked if he trusted May, McDonnell said: "No, sorry".
But the prospect of the government compromising and accepting Labour's demand for close economic ties with the European Union has infuriated pro-Brexit Conservatives, who are demanding May's resignation.
The paper says she could put forward plans for a comprehensive, but temporary, customs arrangement with the European Union that would last until the next general election.More news: How Trump's Hawkish Advisors Won Debate on Iran Oil Sanctions
"I think the basic rate, taking a penny off that, would be talking to the people who need to know we are on their side", he said.
"We can not go on like this, the only way we will leave the European Union realistically is with a leader who actually believes in it", Mr Francois told LBC radio.
More than 100 opposition MPs, including 66 from Labour, warned on Sunday that they would not tolerate a "Westminster stitch-up" on a Brexit deal without a second referendum.
Elsewhere, a poll by political blog Conservative Home found that 82% of its party member panel respondents want Mrs May to stand down as party leader and to call a leadership election.
A temporary customs union would also be likely to raise EU concerns that it could lead to customs checks on the border between euro zone member Ireland and the United Kingdom province of Northern Ireland if it later breaks down - something Ireland objects to strongly.
Shadow chancellor John McDonnell accused Theresa May of jeopardising the cross-party negotiations for her own "personal protection", saying he had lost trust in her and claimed the PM had "blown the confidentiality" of the talks.
"But I think if he wants to do it will actually be surprisingly easy to do because our positions are very, very close. I can't think that any other party in the United Kingdom has raised money like that".
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.