Chris McEleny, the general secretary of the Alba Party, told Sky News he had filed a complaint and asked for an investigation into whether Mr Sunak may have committed contempt of court by mentioning Scotland's former first minister, who is subject to a live police investigation.

The prime minister made a joke about Ms Sturgeon in his speech to Tory Party members at their annual conference in Manchester on Wednesday.

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Ms Sturgeon, who stood down as first minister in February, was arrested and questioned in June as part of Police Scotland's investigation into the SNP's finances but was later released without charge.

She has strenuously denied any wrongdoing.

Mr McEleny's complaint read: "The prime minister is commenting on, and making an assumption about a live Police Scotland investigation.

"Operation Branchform is investigating serious matters of the upmost importance to the people of Scotland.

"It is too important a matter to allow interference from the prime minister in this act of contempt when many people await the facts of Police Scotland's investigation.

"Operation Branchform should be free to pursue its investigation fearlessly without interference from Rishi Sunak."

A Police Scotland spokesperson told Sky News: "We have received a complaint and, following consultation with the Crown Office and Procurator Fiscal Service, no police action is being taken at this time."

Police Scotland launched Operation Branchform to investigate the whereabouts of £600,000 of funding that had been earmarked for a second independence vote.

It is understood there have been complaints the ring-fenced cash may have been used improperly by being spent elsewhere.

Ms Sturgeon's arrest came on the back of the detentions of party treasurer Colin Beattie and former party chief executive Peter Murrell - also Ms Sturgeon's husband - who were both subsequently released without charge.

Police previously searched the SNP headquarters and the home of Ms Sturgeon and Mr Murrell as part of the investigation.

Sky News has approached Downing Street for comment.