Former education secretary Nicky Morgan says Theresa May should consider standing down by autumn 2018 to allow a successor to sell the Brexit deal.
Mrs Morgan was sacked by the prime minister last year.
Shortly after the election, she said Mrs May should not lead the Tories into the next general election.
Mrs Morgan now says the Conservative party “must not miss the opportunity” to think about its next leader once the Brexit deal is finalised.
Michel Barnier, the EU’s chief negotiator, has set a target date of October 2018 for a Brexit deal.
The former cabinet minister was speaking to BBC Newsnight as part of a report into Theresa May’s future as prime minister after the vote on the Queen’s Speech on Thursday.
A deal with Northern Ireland’s Democratic Unionist Party (DUP) means that the prime minister is all but assured of success in the Commons vote.
This will allow the prime minister to press on with meeting her goal of remaining in Downing Street at least until the conclusion of the Article 50 negotiations at the end of March 2019.
But Mrs Morgan says she expects the prime minister may have to leave Downing Street around five months earlier in October 2018.
“Once that shape of Brexit is concluded, once those deals are very much on the table, the Conservative party must not miss the opportunity at that stage to think about who we want to be our future leader,” the former education secretary told Newsnight.
Asked whether that means the prime minister should think of standing down in the autumn of next year, on the grounds that the EU has set October 2018 as the target date for a deal, Mrs Morgan said: “That is probably right – certainly one timetable.”
“Of course one of the things the last couple of years has shown is that making predictions about British politics or international politics is incredibly difficult at the moment.
“But I think the Conservative party – having started on the Brexit road – really is going to own the negotiations, is going to own the shape of Brexit. That is clearly going to be something that will be, if not the issue of the election, will be something that we will be standing on that record in terms of the party going into the next election.”
Downing Street is hoping that success for the prime minister in the Queen’s Speech vote – which is effectively a vote of confidence in government – will strengthen her authority.
But senior Tories have told Newsnight that success could give critics more freedom to speak out because passing the Queen’s Speech entrenches the Tory – but not necessarily Theresa May’s – hold on Downing Street. They say that until the speech is passed, the Queen would be obliged to call on Jeremy Corbyn to try and form a government if Mrs May fell.
Senior Tories have told Newsnight that the prime minister’s future will depend on answers to three questions.
Firstly, is there a credible alternative to Theresa May? Secondly, can the prime minister succeed in the Brexit negotiations? And thirdly, does she have the political strength to rise as a phoenix?
Leading Brexit supporter Jacob Rees-Mogg believes Theresa May could confound her critics.
“With Mrs May it is very hard to tell. But she could be there longer than people are currently speculating. With the DUP there is the basis of a parliamentary majority. Tories MPs don’t want an election, the DUP doesn’t want an election, a lot of backbench Labour MPs don’t much want an election either.”