Asked about the reports, Dame Cressida told a caller on LBC Radio: “You asked if we are investigating. The answer to that is ‘no’. And, as far as I’m aware, we have had no complaints and therefore I really can’t comment on what did or didn’t happen there.”

Asked what would happen if she received a letter, she said: “If I get a letter, I’ll read a letter. This is the Met. We are professional. We are impartial. We act without fear or favour. We follow evidence. That’s what we do.” 

Within minutes of her comments, two Labour MPs announced that they had written to Dame Cressida to urge her to launch an investigation.

They included Barry Gardiner, the former shadow international trade secretary, who claimed he was “surprised” that the Met was “not currently investigating”, adding: “If these events did take place, it implies there is one rule for the Government and another for everyone else.

“I believe your officers should investigate the matter further to establish the facts to see if any laws might have been broken.”

In a separate letter, Neil Coyle, the Labour MP for Bermondsey and Old Southwark, said: “The Prime Minister and his official spokespeople have failed to provide full information about the nature of the gatherings.

“The opaqueness is adding to public mistrust in the regulations which are supposed to protect us all. You know much better than anyone how dangerous mistrust in the system is, and I hope your team will investigate and ascertain the full facts.”