If she chose to fight, she would need the support of more than 50% of Conservative MPs - now 159 - in the confidence vote to stay in office.
May must now quickly seek European Union support for her proposal.
"Nine ministers have now resigned from the two key Brexit departments since Article 50 was triggered".
She said she and the two departed ministers "do not agree about the best way of delivering our shared commitment to honouring the result of the referendum" in which United Kingdom voters opted to leave the EU.
Conservative MPs will then hold a series of ballots to whittle the list of contenders down to two, with the last place candidate dropping out in each round.
"In my view the inevitable effect of the proposed policies will be to make the supposed control by Parliament illusory rather than real". But leading pro-Brexit lawmaker Jacob Rees-Mogg said he didn't think she would face a leadership challenge.
May's proposals for a future European Union relationship after Britain departs from the bloc next March had taken two years of internal wrangling to agree.
Johnson's departure came a day after David Davis, the British minister in charge of Brexit negotiations, also stepped down in protest of May's new Brexit strategy that pointed to a more "business-friendly" deal with the EU.
Will the resignations weaken or strengthen her hand?
The prime minister's official spokesman said she would fight any attempt to oust her as prime minister if the required 48 Tory MPs called for a contest.
Dominic Raab, a Brexit supporter and former housing minister, was appointed to replace Davis only days before negotiations in Brussels are due to resume next week.
In his resignation letter, Johnson harshly criticised the government's approach to Brexit: "Brexit should be about opportunity and hope".
Under her proposal, a treaty would be signed committing the UK to "continued harmonisation" with EU rules - avoiding friction at the UK-EU border, including Northern Ireland.
Britain and the European Union hope to reach broad agreement by October so the national parliaments of the remaining countries can ratify a deal before Britain leaves. "This is the right Brexit", she said.
Opposition Labour Party leader Jeremy Corbyn said the government was incapable of delivering Brexit.
"I have no idea, Sir Graham Brady has made it very clear that he will not give anybody any indication of how many letters he has received", he said.
While the Irish government dismissed the resignations as important, saying they would continue to negotiate with Mrs May herself, Mr Coveney's comments will merely add to Mrs May's growing Brexit headache.
"In a vote of confidence in the House of Commons I would support the Prime Minister that's because I don't want to have another General Election, there isn't a vote of confidence in either case at the moment, I don't think many people want another election, we only had one a year ago".
That would pair arguably the two best players in the league with one of the NBA's most storied franchises. LeBron James could be leaving Cleveland, and TUMS wants to help Cavs fans deal with the potential loss.
Screen Time will help users keep track of their smartphone usage - and even set limits on app usage across certain categories. Users who sign up on for the free Apple Beta Software Program can gain access to iOS 12 beta 1 and tvOS 12 beta 1.
In 2013, Jackson appeared on CNN show Piers Morgan Tonight and claimed to have no regrets about the way he raised his children. Just this past weekend, Janet Jackson hailed her father during an acceptance speech at a Radio Disney awards ceremony.
Many haven't had a chance to take the gender-segregated driving courses that were first offered to women only a few months ago. Saudia Arabia's transformation, including allowing women greater freedoms and equality, may be essential to its economy.
Last year's victor , a delightfully floppy and seemingly half-melted Neapolitan Mastiff named Martha, didn't compete this year. Some of this year's contestants had hairless bodies, others had lolling tongues.
The company will take an active role in helping interested entrepreneurs start, set up and manage their own delivery business. Amazon is getting into the package delivery business, with Youngstown one of many markets the company is targeting.
For the second time this year, actress Heather Locklear has been arrested for battery on a police officer and emergency personnel. Still, the source says Locklear's friends and family are hopeful that she will take her latest expected stint in rehab seriously.
The world number one was preparing for his Wimbledon second round match against Kazakhstan's Mikhail Kukushkin on Centre Court . Guido Pella of Argentina took out last year's beaten finalist Marin Cilic of Croatia in the second round at Wimbledon.
Apart from the colourful lines it is hard to tell where the hell you are most of the time which is the part that I dislike. A wheel then came off Sergio Perez's Force India vehicle at speed on the Mistral straight in the late afternoon.
Still, Ice Universe's claim runs directly opposite the increasing number of reports surrounding Samsung's upcoming launch plans. We do not have anything better to offer and companies rarely use logic when naming various phone models so it may well happen.
Despite a 75 in round two, Korda was still able to climb the leaderboard on Sunday with a 68 for her fifth top 10 of the year. Angel Yin made seven birdies on Sunday and shot just 1-under and tied for fourth, her best finish in a major championship.
Joining Dhoni on the sidelines were Shikhar Dhawan and, as expected, the two opening bowlers Jasprit Bumrah and Bhuvneshwar Kumar. Perhaps it was a marker of this crucial impending assignment that India chose to stay on and practice in London on Monday.
The move puts more pressure on Sony , which has remained mum on the possibility of further opening up cross network play. Sony has ignored pressure from the public and from game makers regarding opening the PS4 to cross-platform play.
There is guaranteed to be a fourth straight European team winning the World Cup on July 15 in Moscow. And De Bruyne feels Brazil's inability to properly adapt to Belgium's tactics ultimately cost them.
But by introducing Bluetooth features, Samsung has the potential to make it a phenomenal addition to the Galaxy Note line. We already know the Note 9 will be announced on 9 August in NY (we've collected all the rumours on that over here).
Most recently, his stepson Riza Aziz was also called into the MACC's headquarters today for questioning relating to the scandal. It follows an investigation into how billions of dollars went missing from a state fund - which he founded a decade ago.
By brawling with America's friends, critics say , Trump has squandered an opportunity to build a united front against China. Traders are reacting to retaliatory events and the imposition of fresh tariffs at the end of the week.
Abdullah reportedly said that unveiling the plan without coordinating with the Palestinians would be a "catastrophe". Abbas appealed to the Prince and the British people to support the Palestinian campaign for independence.
Belgium have avoided another World Cup shock after coming from behind to beat Japan 3-2 in a thrilling last-16 clash. De Bruyne advanced the ball with a handsome push, making a flawless lead pass to Jan Vertonghen with room to run.
The president reversed the family separation policy last week but the re-unification of separated families remains an issue. "I'm here to help and support you and give my help, whatever I can, on behalf of children and the families", she added.
England manager Gareth Southgate did not confirm or deny any team news at his press conference but was full of praise for Kane. Harry Kane's involvement is one of the tricky decisions facing Southgate.