His Tweet was seen as a lukewarm message of backing.

Ms Gray is the civil servant leading the probe into allegations of lockdown-breaking parties being held in Downing Street.

Mr Sunak is yet to appear on radio or television giving backing to Mr Johnson, unlike Liz Truss, the Foreign Secretary, who is also expected to run should a Tory leadership race emerge.

Other Cabinet ministers and Tory whips have warned potential successors against moves to build up support as Mr Johnson tries to address the political crisis he faces.

In a warning to those considering a push for the leadership, one Cabinet minister told The Telegraph: “Be careful what you wish for.

“History is littered with examples of prime ministerial hopefuls who’ve got their timing wrong and they were never heard of again.”

A Tory MP who is part of the whipping operation, which is in charge of party discipline, said of leadership rivals: “They would be well advised to keep a low profile. Anything they do now would be counterproductive.”

Advisers to both Mr Sunak and Ms Truss have denied their bosses are on leadership manoeuvres and insist they remain loyal to Mr Johnson.

But their meeting and socialising with Tory MPs over recent months have been viewed with suspicion by the Tory whips.

Ms Truss’s drinks events with Tory MPs have been “Fizz with Liz” and her meetings with donors “Biz with Liz”, while Mr Sunak often meets MPs to discuss Treasury policy.

Ms Truss, who came top in a recent poll of Tory members about who should be the next party leader, has been more publicly supportive of Mr Johnson than Mr Sunak since the Prime Minister’s apology on Wednesday over allegations of lockdown-breaking parties.

Ms Truss tweeted that day: “The Prime Minister is delivering for Britain – from Brexit to the booster programme to economic growth. I stand behind the Prime Minister 100% as he takes our country forward.”