Conservative MP Mark Francois has called for an “indicative vote of confidence” on Theresa May’s leadership this week.
The prominent Eurosceptic said the Prime Minister should resign for the sake of the “existential future of our party and the destiny of our country”.
In a letter to the chair of the backbench Tory 1922 Committee, Sir Graham Brady, Mr Francois said an “indicative vote” would send a signal to EU leaders that May did not command the support of her party.
A formal vote of confidence in Mrs May as Conservative leader cannot be held until December, after she survived an earlier attempt to oust her by 200 votes to 117, granting her a 12-month period of grace during which no challenge is permitted.
Mr Francois wrote: "We simply cannot go on like this, with a weak leader, a riven Cabinet and a party in despair.
"I believe Theresa May has been a failure as leader of our party, which she now threatens to destroy. Hers is a classic example of hubris - and after hubris comes nemesis."
Sir Graham told last week's meeting of the 1922 executive that a proposal for an indicative vote had been put to him.
The executive decided that, while it was possible for such a vote to be staged, it was neither necessary nor appropriate to do so at this point.
Mr Francois accused a “coterie of neo-federalist civil servants” and a “powerful Remainiac cabal in the Cabinet” of attempting to “lock” the UK into a customs union.
Meanwhile, he accused Brexiteers in the Cabinet of doing "absolutely nothing save consume vast quantities of pizza" as "riven by personal ambition, they keep their heads firmly below the parapet, many of them hoping that by sitting out this great battle for the destiny of our country, they will somehow inherit the crown".
The former shadow Europe minister faced criticism last week after his use of a Bible quotation as part of a Brexit debate was branded as “deeply offensive” to Christians.