Theresa May is expected to announce the date of her departure from Downing Street on Friday morning, senior cabinet ministers have said.
Sources say they expect the PM to give a timetable for her successor to be chosen, with 10 June likely to be the start of the official leadership race.
Mrs May is due to meet the chairman of Tory backbenchers on Friday.
She has been under pressure to resign, after a backlash by her own MPs against her latest Brexit plan.
Since January, Parliament has rejected the withdrawal agreement Mrs May negotiated with the EU three times. Recent attempts to find a formal compromise with Labour also failed.
Mrs May had planned to publish the Withdrawal Agreement Bill on Friday - the legislation required to bring the agreement into UK law - describing it as "one last chance" to deliver Brexit.
However, her proposals - including a customs union arrangement and an offer to give MPs a vote on holding another referendum - angered many Tories.
Labour said it was a "rehash" and they would not support the plans.
Andrea Leadsom quit as Commons leader on Wednesday evening saying she no longer believed the government's approach would "deliver on the referendum result".
And on Thursday, Mrs May met Home Secretary Sajid Javid and Foreign Secretary Jeremy Hunt at Downing Street, where they are understood to have expressed their concerns about the bill.
BBC political editor Laura Kuenssberg says several cabinet ministers have said they expect Mrs May to announce her departure on Friday morning, although there has been no confirmation from Downing Street.
Ministers hope the campaign for the next Conservative Party leader can be finished by the end of July.
Unless something extremely strange happens in the next couple of days, it is now, really, nearly over.
Why now though? It's not as if Theresa May's been having an easy time of it for months.
You guessed it, it's Brexit, and what's accelerated her departure was trying - again - to put her Brexit plans to Parliament.
A leadership contest beginning on 10 June would come after the state visit from US President Donald Trump and the Peterborough by-election.
Sir Graham Brady, the chairman of the 1922 Committee of Conservative backbenchers, was due to meet the prime minister on Friday morning.
He was thought to expect her to name a date for her departure - with the possibility of further moves by Conservative MPs to instigate a vote of no confidence in her, if she did not.
On Wednesday, members of the committee held a secret ballot on whether to change party rules, to allow the prime minister to face a vote of no confidence immediately.
Under current rules, she cannot face another confidence vote within 12 months of the last one, which was in December 2018.
However, the result of the secret ballot will not be disclosed if Mrs May sets a date for her departure.
What are the Tory leadership rules?
On Thursday, the UK voted in the European Parliamentary elections, two months after it was originally due to leave the European Union.
Results will not begin to be announced until Sunday night, after voting concludes across the EU.
The prime minister had already pledged to set a timetable for a new leader to take over, once MPs have voted on the Withdrawal Agreement Bill.
But calls for her to leave sooner began to come from Conservatives who had, until now, stayed loyal.