Virendra Sharma, a supporter of the Best for Britain campaign for another referendum, said that a "gang of Brexit bullies are threatening to pull the plug on the Prime Minister".
May is holding a meeting of her new-look cabinet on Tuesday, following a forced reshuffle in the aftermath of the resignations.
The resignations rocked May in a week that includes a North Atlantic Treaty Organisation summit starting Wednesday and a United Kingdom visit by US President Donald Trump from Thursday.
But leading pro-Brexit lawmaker Jacob Rees-Mogg said he didn't think she would face a leadership challenge.
The trans-Atlantic relationship has had some awkward moments since Trump's election.
Johnson criticised the Brexit blueprint in private but has so far refrained from public comment.
Ministers gathered for a retreat at the premier's country residence, Chequers, on Friday for talks called at short notice, and mobile phones had to be left outside.
On Tuesday, the president described Johnson as a "friend" who had been "very nice to me, very supportive", even though when Trump suggested in 2015 London had police no-go zones, Johnson, then the capital's mayor, quipped he feared going to NY because of "the real risk of meeting Donald Trump".More news: Hurricane Chris intensifies in Atlantic, takes aim at Avalon Peninsula
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British Prime Minister Theresa May dug in her heels today after the resignation of two top government ministers over Brexit negotiations whipped up a storm that threatened to topple her fragile minority government.
Above a picture of her cabinet, including her new foreign secretary, Jeremy Hunt, and Brexit minister Dominic Raab, May tweeted: "Productive Cabinet meeting this morning - looking ahead to a busy week".
In his resignation letter, Johnson wrote: "Brexit should be about opportunity and hope".
The businesswoman and activist said: "You should never have given a very complex question to the people of Britain".
Under Conservative Party rules, a confidence vote in a leader can be triggered if 15 per cent of Conservative lawmakers - now 48 - write a letter requesting one. "It is the last thing we need".
In the immediate aftermath of the Cabinet signing up to Theresa May's soft Brexit strategy on Friday, the odds of a Conservative leadership contest in 2018 were 2/1.
This means that Michael Gove remains unmoved from his position heading up the Department of Environment, Food and Rural Affairs (Defra).
The UK is due to leave the European Union on 29 March 2019, but the two sides have yet to agree how trade will work between the UK and the EU after that.
"Being tied to European Union regulations and the European Union tying our hands when seeking to make new trade agreements will be the worst of all worlds", wrote the Mansfield MP, who voted remain in a constituency where more than 70 per cent of voters opted to leave.