Over the weekend, Mr Sunak ordered a leak inquiry to find and possibly prosecute the culprit who disclosed his wife’s non-dom status, which had allowed her to potentially avoid millions of pounds in UK tax.

Backed by the Cabinet Office, it is set to examine not only the “keepers” of the sensitive information and those who had access to it but “anyone” who requested to see it.

“The inquiry will be carried out with a view to potential criminal prosecutions because it is against the law to leak someone’s tax status,” said a source.

However, senior Tories warned that a leak inquiry could be counterproductive by keeping the story going in the run-up to the local elections in May.

Sir Iain Duncan Smith, the former Tory leader, said: “It is a mistake because no leak inquiry ever discovered the source but what it does do is keep the story running. It makes it look like he wants to hide it. It sounds worse than it is.”

A Cabinet minister said it was “a complete waste of time” and that Mr Sunak should let matters lie. They said: “If he feels it’s necessary, that’s fine but I wouldn’t have done it myself. I tend to stick with ‘don’t explain, don’t complain’ when it comes to these things. The leak inquiry is a complete waste of time.”

A former Cabinet minister said: “The surprising thing is that he didn’t regularise this before he became a minister or went into the Treasury. It is a sign of his inexperience.

“He is right to ask how did this all happen but to make a song and dance with a leak inquiry isn’t the right way to go about these things. It just keeps the story going.”

Other MPs said that the issue had not yet gained traction on the doorsteps, raising questions over whether it would be wiser to let the issue lie.

Government officials refused to identify the legislation under which any potential leaker would be prosecuted. One option could be data protection laws, which make it a criminal offence to disclose “knowingly or recklessly” personal data without consent.

A senior ally of the Chancellor criticised those out to “bring down people who are capable in Parliament”. He suggested leadership rivals were seeking to “damage the person who is one of the main contenders” for the top job. “I see envy and thinly veiled racism around some of this,” he said.

In the wake of the focus on his wealth, Mr Sunak is to embark on a “town hall tour” of the country to counter claims he is out of touch.