As many as two-thirds of Labour MPs, including several shadow cabinet ministers and frontbenchers, have reportedly said they will refuse to back a deal without a confirmatory referendum.
However, shadow chancellor John McDonnell said he had no trust in Mrs May after reports emerged that she is ready to offer a temporary customs arrangement with the European Union to the opposition.
Such is the scale of the electoral disaster facing the party, however, and the level of dissatisfaction with Mrs May's leadership, that the '22 is considering amending its rules to allow her to be removed earlier.
It comes as the ruling Conservatives and opposition Labour continue talks aimed at breaking the deadlock around PM May's Brexit deal.
He made the claim as Theresa May's de facto deputy - David Lidington - appearing to admit defeat in avoiding the May 23 poll. "We now wish to see the withdrawal agreement introduced, and ratified by parliament, as soon as possible and ideally before MEPs take their seats".
She was to meet the chairman of the Conservative 1922 Committee, an influential party group which can make or break Conservative Party leaders, on Tuesday.
"It would be much easier and I think the European elections would be much easier if she did set out her own timetable to go but it is up to her".
Party leader Jeremy Corbyn said Brexit had a big impact on last week's local election results, which saw huge losses for the Tories and a disappointing showing for Labour.
The Tory party lost a staggering 1,334 council positions as the public seemingly punished both main parties for failing to deal with Brexit.
Labour sources were dismissive of weekend reports the government would offer new concessions, including a temporary customs union with the EU until a national election due in June 2022, saying such a proposal would not go far enough.
"I want to look at whatever deal is come to between the parties and I know this is a crucial week", he told BBC radio.