Ms Reeves previously admitted to breaching local council housing rules by failing to secure a “selective” licence for the property when it was rented out following her move into No 11 Downing Street.

She initially said that she was unaware of the requirement, but emails between her husband and the letting agency published on Thursday showed he had been informed about the need for a licence.

But estate agents Harvey and Wheeler took the blame for the “oversight” in not applying for a licence on her behalf, despite having agreed to do so.

Conservative Party chairman Kevin Hollinrake told Sky News yesterday (FRI) that he is not satisfied with the resolution of the scandal.

He said: “What we’re calling for, and I think what Kemi Badenoch was calling for, is absolutely right – a full investigation.

“And that’s within the prime minister’s gift, to show some backbone and make sure all these events are properly investigated.

“And if it turns out it’s an innocent mistake, of course, that’s not a resigning matter, but if it’s more serious than that, then it potentially is.”

He went on: “The point is that on Wednesday, she said she wasn’t aware. On Thursday, she said she was aware, but it was the agent’s fault.

“That story has changed and I think Rachel Reeves, her problem here is the hypocrisy.

“She had in the past, when this has happened to other people in other parties, she’d call for their resignation. We’re not calling for that. We’re calling for an investigation. I think it’s perfectly in order that we should do that.”

The Prime Minister said in a letter to Ms Reeves, published on Thursday night, that after reviewing the correspondence, “I still regard this as a case of an inadvertent failure to secure the appropriate licence, which you have apologised for and are now rectifying”.

He added: “Having consulted the Independent Adviser on Ministerial Standards and received his clear and swift advice, with which I concur, I see no need for any further action.”

Sir Keir said it was “regrettable” that she had not shared the information in her first letter to him about the matter on Wednesday, but said he accepted she was “acting in good faith”.

“It would clearly have been better if you and your husband had conducted a full trawl through all email correspondence with the estate agency before writing to me yesterday,” the Labour leader said.

The Prime Minister’s independent standards adviser, Sir Laurie Magnus, told him that after seeing the emails, he still believed Ms Reeves made an “unfortunate but inadvertent error”.

In a letter to Sir Keir, he said he found “no evidence of bad faith” in Ms Reeves’s contradicting statements about her awareness of the need for the licence.