The Prime Minister is hoping her much-maligned Withdrawal Agreement will finally be passed this month, following talks with Labour to break the deadlock.
Sir Graham Brady, the chairman of the Conservative backbench 1922 Committee, suggested Mrs May could make a final effort to get her Brexit deal through Parliament before European elections on May 23 - where the party is expected to suffer heavy losses.
He said he expected ministers to bring forward the Withdrawal Agreement Bill - required to ratify her deal in law - in the "near future”.
"The executive is very keen to meet the Prime Minister and will have a full opportunity to discuss and to reach whatever conclusions it wishes to reach next week," Sir Graham said.
"It's my understanding it's the Government's intention to bring a second reading of the Bill forward in the near future, certainly the intention is before the European election takes place.”
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Justice Secretary David Gauke also confirmed the plans for another vote, telling BBC Radio 4’s Today programme: “We want to bring it [the withdrawal agreement] back as quickly as we can and we want to succeed… we shouldn’t give up.”
However, Mrs May has a fight on her hand to get the bill passed as she has to appease Eurosceptic Tories who are already calling on her to quit after the local council elections drubbing.
A deal struck with Labour would also prove unpopular with Brexit-supporting Tory backbenchers.
Despite the pressure on the PM to quit and let someone else take over Brexit, she rebuffed fresh demands to set out a timetable for her departure from No 10 in a meeting with Sir Graham on Wednesday.
He said Mrs May would meet the committee's 18-strong executive next week.
A No 10 source said: "The PM made a very generous and bold offer to the 1922 Committee a few weeks ago that she would see through phase one of the Brexit process and she would leave and open up for new leadership for phase two.
"That's the timetable she is working towards. She wants to get Brexit done.”
Meanwhile, talks between the Government and Labour on a compromise deal to end the deadlock are set to resume next week, following a second day of "extended" discussions in Whitehall.
A No 10 spokesman said the latest meeting "demonstrates the seriousness with which both sides are approaching these talks”.
A Labour Party spokesman added: "After the second day of talks this week, the negotiating teams are working to establish scope for agreement and will meet again at the beginning of next week.”