Nigel Adams, a former Minister who quit last week over May's decision to hold talks on Brexit with opposition Labour leader, Jeremy Corbyn, said: "Over 170 Conservative MPs including cabinet ministers signed a letter to the Prime Minister last week urging her to ensure the United Kingdom does not take part in the European elections".
"It would be, I think, a suicide note of the Conservative Party if we had to fight the European elections", he told BBC Radio 4's Today.
She will head to an European Union summit Wednesday seeking to secure another Brexit extension, until June 30, to prevent Britain crashing out the bloc at the end of next week with no deal.
Talks between the government and Labour broke down on Friday, with the opposition saying it was "disappointed" with the concessions ministers were willing to offer.
"We need to get a deal over the line and that's why we've been looking for new ways to find an agreement in parliament - and that means cross-party talks", May said in a video recorded at her country retreat Chequers. But we should be open to listen to suggestions that others have made.
Ahead of a week in which she will go to Brussels seeking to secure a further delay to the UK's departure, the Prime Minister said: "The longer this takes, the greater the risk of the United Kingdom never leaving at all".
In a statement released late Saturday, May admitted Brexit could only be delivered with support from the opposition Labour Party.
Shadow home secretary Diane Abbott said: "We are engaged in these talks in good faith but the Government perhaps has to show a little more flexibility than it seems to have done so far".
He said: "At a time when Conservative members and activists are recoiling in horror at the idea of collaborating with Jeremy Corbyn, the No 10 effort to rehabilitate this terrible deal and worse strategy to drive it through seems wholly forlorn".
The discussions' "overall mood is quite a positive and hopeful one" despite the government's "disappointing" failure to shift its stance on several issues, she said. A Theresa May has repeatedly said Britain can not be in a customs union but has argued that her deal and the Northern Ireland...
"I understand that European Union colleagues are somewhat fed up that the process has taken as long as it has; we are also fed up that we haven't been able to complete this earlier, but I am very confident that we will get it done".
The UK has already sought one extension to the March 29 Brexit deadline and is set to crash out of the bloc on April 12 unless an alternative arrangement is agreed with the EU.
However, it remains to be seen what the remaining European Union member countries will be willing to agree to, with any decision on an extension to the Brexit mechanism requiring their unanimous backing.
May badly needs a breakthrough in London before she travels to Brussels on Wednesday.