Chris Bryant, the chair of the committee on standards, has called for an independent investigation into the alleged Christmas parties held across Whitehall and insisted ultimately it should be led by the police. The cabinet secretary, Simon Case, on Friday night quit his role leading the investigation into whether three alleged Whitehall parties breached Covid restrictions after it emerged an event was held in his private office last December. Rules in London at the time banned indoor social gatherings. Sue Gray, a top civil servant in the Department for Levelling up, Housing and Communities, has taken over Case’s role to investigate the alleged gatherings – two of which are said to have taken place in Downing Street on 27 November and 18 December and another at the education department on 10 December. Bryant told BBC Radio 4’s Today programme on Saturday: “In the end, I think it should be the police investigating. “But maybe the first stage of getting all the facts and figures together needs to be done by a completely independent figure, not somebody who is in the midst.” In a separate interview with BBC Breakfast, the Labour MP said that “if the rules are broken … in any other line of work, the police would be investigating and I don’t know why the police aren’t investigating this situation. “In the end, the final analysis has to be done by [a] completely independent person. I think that that should be the police.” Any evidence of lawbreaking, he added, should be handed over to the police. Bryant also said there was a “good argument” for the chair of the committee on standards in public life, Jonathan Evans, to take on the role of leading the investigation. “I think quite a lot of people had a bit of anxiety anyway about Simon Case taking on the job [of the investigation] because it felt a bit like getting the housekeeper to investigate whether the housekeeper has had a party,” he said. Bryant said Gray was likely to know “every single one” of the people she will be investigating, adding: “I’m not convinced that this is the right direction. And this is the kind of problem that you have with a completely dysfunctional Downing Street.” The Scottish National party’s Westminster leader, Ian Blackford, also said the investigation should be led by someone outside Whitehall, adding “having somebody else from the civil service marking their own isn’t good enough”. He told the BBC’s Newsnight there should be a “judge-led inquiry”. Meanwhile, Gavin Barwell, the former Downing Street chief of staff to the then prime minister, Theresa May, backed Gray for the role. “I can’t think from my time in government of a better person to put in charge of this review than Sue Gray,” he told the BBC. “She had the role in charge of propriety and ethics in government when I was chief of staff. “I saw her handle a number of investigations in a way that was completely independent and her determination always to get to the truth and present the prime minister with all of the evidence.” He added: “I think that she is exactly the right kind of person and we can be confident that we will get the facts.”