Brexit-supporting members of the committee are pushing for a contest to take place as soon as possible, with MPs agreeing a shortlist of two candidates before the summer recess and then Tory activists voting over the holiday period. The National Conservative Convention's vote on 15 June is non-binding, though, so the PM could ignore it.
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.
"Our policy position has not changed since the last Labour conference, where a public vote was one of many options on the table, certainly to avoid a Tory deal, a bad Tory Brexit or a no-deal situation".
Lord Heseltine's comment are yet another blow to Mrs May as not only has she admitted failure in avoiding the European Union elections but she has also seen her party suffer a colossal defeat in last week's local elections.
Referring to talks with Labour, children's minister Nadhim Zahawi said: "Currently, if you look at the Withdrawal Agreement, the customs arrangements, or the alignment with the European Union, would go on until December 2021, if Labour believe, actually, that they would rather go to the next general election, which is 2022, for example, then, actually, that's still a temporary customs arrangement".
However, while Labour will want to string out the talks, they will not want to join what Mr Farage has already branded "a coalition against the people" to help Mrs May's deal through. Reality Check unpacks the basics.
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Because to do so would split her party down the middle, and cause maximum chaos for her.
The Government has accepted it can not get its Brexit deal through Parliament in time to avoid European elections on May 23.
It was triggered after 65 local Conservative associations said they had lost trust in the prime minister.
Will the cross-party talks get anywhere this week?
May is preparing to make new proposals of a temporary customs union at least until the next election in 2022.
A senior government source says it IS possible, though, to see a way to a deal, but it is unlikely to be resolved this week - and their aim is not to create some kind of May-Corbyn Rose Garden moment (imagine!) but to set out a path to get the Withdrawal Bill to Commons with a fair wind.