The prime minister has been reported to the police by Labour MPs over allegations there were at least two parties in Downing Street during lockdown restrictions last year.

Neil Coyle, Labour MP for Bermondsey and Old Southwark, wrote to the Met police commissioner, Cressida Dick, asking her to investigate reports that the prime minister spoke at a leaving do in November and also allowed a staff Christmas party to go ahead in December.

At the time, there were lockdown restrictions in place preventing socialising. After sending the letter, Coyle said: “I believe they broke the law. Most of my constituents followed the rules; those that didn’t faced penalties. Johnson is not above the law, despite his bloated self-entitlement.”

Barry Gardiner, the former shadow international trade secretary, also asked the Met to investigate, adding: “If these events did take place, it implies there is one rule for the government and another for everyone else.

Johnson has not explicitly denied the parties took place but No 10 has insisted “all Covid rules were followed”. Earlier on Friday, the Metropolitan police had said it had not yet received any complaints about the parties.

Separately, the former regional chief prosecutor, Nazir Afzal, called for the forthcoming public inquiry into the pandemic to examine alleged lockdown breaches at the two Downing Street parties last year as well as Dominic Cummings’s journeys to the north-east.

Afzal wants the inquiry, which is due to start in the new year, to examine how compliant officials were with the government’s own rules and guidance and the impact of the alleged breaches on public compliance.

In a letter sent to the prime minister on Friday, Afzal’s lawyers said the inquiry should have “full authority to explore the role of individuals in government, and those connected to the government, in devising the regulations, complying with them themselves and, in turn, ensuring the public’s confidence and compliance with those regulations, as well as the impact of non-compliance”.

The letter cited Afzal’s concerns about Cummings travelling to Durham and Barnard Castle at the height of the first lockdown when he was chief aide to the prime minister.

It added: “This was not the only example of non-compliance to emerge at the time and, in the light of recent revelations about a party held at Downing Street last Christmas, it is likely that more will emerge.”

It comes as the Conservatives said they were pressing ahead with this year’s Christmas party in spite of scientists’ fears over the spread of Omicron, as their co-chair told people to “keep calm and carry on” with festivities.

“We obviously wouldn’t set out details of private functions in No 10 but, as I say, there will be festive events in the run-up to Christmas,” said Oliver Dowden.

He told Sky News: “The message to people, I think, is fairly straightforward – which is: keep calm, carry on with your Christmas plans. We’ve put the necessary restrictions in place, but beyond that keep calm and carry on.”

Asked about the Tories’ own party plans, he told BBC Breakfast: “I think it is still planned, and I don’t intend to cancel it as chairman of the Conservative party.”

On a trip to North Shropshire, Johnson echoed his chairman, saying there was no need for cancellations. “On the subject of Christmas parties, I’ve noticed there’s been quite a lot of to-ing and fro-ing about it, people concerned that they need to cancel their Christmas parties. That’s not right, we’re not saying that and we’re not saying that nativity plays have to be cancelled,” he said.

Labour has decided to cancel its Christmas function though it is not urging businesses to do the same.

The latest Covid restrictions call for mask-wearing in shops and on public transport but there are no restrictions on socialising, unlike last year before the vaccine programme had been rolled out.