Theresa May has promised to set a timetable for the election of her successor after the next Brexit vote in the first week of June.
The agreement follows a meeting between the prime minister and senior Tory MPs who are demanding a date for her departure from Downing Street.
If she loses the vote on her Brexit plan, already rejected three times, sources tell the BBC she would resign.
Meanwhile, Boris Johnson has said he will run for leader once Mrs May goes.
Mrs May survived a confidence vote of Conservative MPs at the end of last year and existing Conservative rules mean she cannot formally be challenged again until December.
But the prime minister has come under increasing pressure to leave Downing Street this summer, amid the Brexit impasse and poor results for the Conservatives in the recent local elections in England.
The BBC's political editor Laura Kuenssberg said senior sources had told her it was "inconceivable" the prime minister could remain in office if MPs rejected her Brexit plans for a fourth time.
Have your say
More World Headlines
- Netanyahu's wife admits misusing public funds
- Roadside bomb kills several police officers in east Kenya
- Ivanka Trump, Jared Kushner took in at least $135m in 2019
- Hong Kong suspends controversial extradition bill after mass protests
- Outrage in Tanzania over wig and hair extension tax
- Christchuch attack suspect pleads not guilty
- Kenyan MP arrested 'for slapping female colleague'
- German court rules mass-killing of male chicks legal
- Mexico says sale of presidential plane will curb illegal migration
- Gabon appoints white British conservationist as forestry and environment minister
- Public protector confirms Ramaphosa is implicated in Bosasa probe
- Russia has dropped charges against a journalist after protests
- Trump accidentally reveals Mexico migrant plan
- Ebola outbreak spreads to child in Uganda
- Chemical castration: Alabama enacts new paedophile law