Lord Hague warned "Tory MPs with their pens hovering over letters" that the chances of such a challenge leading to the Brexit they desire "are about zero".
Opposition Labour leader Jeremy Corbyn said the Cabinet ministers had "jumped the sinking ship".
I thought yesterday that he was entirely out of line with his comment on the resignation of Boris Johnson as Foreign Minister in the cabinet of British Prime Minister May to protest her betrayal of Brexit.
May said she had chaired a "productive" meeting of her government, unswayed by the resignations of Foreign Secretary Boris Johnson and Brexit negotiator David Davis that rocked the government on Monday.
After an hours-long meeting last week at her Chequers country residence, May seemed to have persuaded the most vocalBrexit campaigners in the cabinet to back her plan to press for "a free trade area for goods" with the European Union and maintain close trade ties.
Yet this was not even the biggest thing Theresa May had to worry about as the U.S. president left the White House for Europe ahead of his visit to Britain, which starts with the North Atlantic Treaty Organisation summit in Brussels today.
The trans-Atlantic relationship has had some awkward moments since Trump's election.
Arriving at the North Atlantic Treaty Organisation summit in Brussels, the Prime Minister said: "It's there because it delivers on the vote that people gave on Brexit, it delivers the fact that we will have an end to free movement, we will have an end to the jurisdiction of the European Court of Justice in the United Kingdom, we won't be sending vast contributions to the EU every year, we'll be out of the Common Agricultural Policy, out of the Common Fisheries Policy. Maybe I'll speak to him when I get over there".
He added: "The stance of hard-line Brexiteers is revealing, Boris Johnson's resignation letter to Theresa May in itself reveals actually the degree the type of Brexit he wanted was never going to happen, and he was never really dealing with reality".
Asked whether May would contest a no-confidence vote, a Downing Street source said simply: "Yes".
Zakharova pointed at the resounding failures of Johnson's foreign policy, including the ongoing Skripal saga and the obsession with discovering of the dreaded "hand of Kremlin" in many issues. But, after much speculation, Mr Johnson announced he would not enter the Conservative leadership race when his ally Michael Gove announced he was running.
Merkel, a key voice in Brussels, said the other 27 European Union members would give a joint response, "but it's good that the proposals are on the table - that much I can say already, without going into details". "There are all sorts of question marks under this new plan", she said. Brexit-supporting lawmakers were angered by the proposals, saying they would keep Britain tethered to the bloc and unable to change its rules to strike new trade deals around the world.
In doing so, the Gibraltar Government noted that Mr Johnson and Mr Davis have both been strong supporters of Gibraltar in Government and outside it.
Markets had welcomed May's proposals as they retain close trade ties with the EU.
Brexiteer Walsall North MP Eddie Hughes is one of a number of Tory backbenchers to have voiced concerns over the detail of Mrs May's Brexit deal, with the full 120-page version not due to be released until Thursday.