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Theresa May has failed to name a date for her departure from Downing Street but vowed to "agree a timetable for the election of a new leader" within weeks.
Senior figures on the 1922 Committee held talks with the prime minister for some 90 minutes in her Commons office, amid pressure on Ms May to declare when she will stand down.
Chairman Sir Graham Brady said he would meet Ms May again in June to "agree a timetable for the election of a new leader" after a make-or-break vote on her key Brexit legislation.
In a statement, he said: "The prime minister is determined to secure our departure from the European Union and is devoting her efforts to securing the second reading of the Withdrawal Agreement Bill (WAB) in the week commencing 3 June 2019 and the passage of that bill and the consequent departure of the United Kingdom from the European Union by the summer.
"We have agreed that she and I will meet following the second reading of the Bill to agree a timetable for the election of a new leader of the Conservative and Unionist Party."
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Sir Graham said there were "frank discussions" between the executive and the prime minister, where it was agreed that he would meet Ms May regardless of whether the bill is backed by MPs.
He told reporters: "We have agreed to meet to decide the timetable for the election of a new leader of the Conservative Party as soon as the second reading has occurred and that will take place regardless of what the vote is on the second reading - whether it passes or whether it fails."
Sir Graham added: "It was a very frank discussion, I tried to make sure that all the views represented on the executive were expressed and we had a very frank exchange with the prime minister."
Downing Street has said that Sir Graham Brady's statement was issued with the consent of the prime minister.
A spokesperson said: "The PM's focus is on securing our departure from the EU and we are working incredibly hard to build a stable majority for the WAB (Withdrawal Agreement Bill) ahead of its introduction.
"That is what we are focused on."
The meeting comes as a result of a promise from Ms May to stand down once the first phase of Brexit had been delivered.
But Tory MPs have become increasingly frustrated by the prime minister's refusal to name a date for her exit from Downing Street, while progress on delivering Brexit has ground to a halt.
In a last attempt to break the deadlock, Ms May pledged to bring the bill to the Commons in the week of June 3.
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She still faces major opposition from both Labour and Tory MPs to get her Brexit blueprint over the line.
The leading Brexiteer insisted that there was currently no vacancy at No10 but his comments will be taken as confirmation he intends to throw his hat into the ring.
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