Downing Street is facing calls to ensure that Boris Johnson will be personally interviewed by the Sue Gray inquiry about alleged No 10 gatherings during the first lockdown, after it emerged he may have been present at a “bring your own booze” party that month.

The inquiry, into allegations of social mixing bans being broken in No 10, was widened this weekend to include reported gatherings from May 2020, amid reports that an official emailed Downing Street staff inviting them to socially distanced drinks.

No 10 did not deny on Sunday that the prime minister and his wife attended the event on 20 May, which is said to have been organised by a senior civil servant in Johnson’s private office, Martin Reynolds, with food and wine set out on tables.

It comes after the Guardian reported a “wine and pizza” party in Downing Street in the garden and inside No 10 on 15 May, with staff drinking late into the evening after a press conference that day. After No 10 insisted staff were working, the Guardian obtained a photograph of the prime minister and his wife sitting with officials at a table with wine and cheese, with 15 other staff in the background and bottles of wine visible.

These events come alongside reports of at least five other potential gatherings in Downing Street and Whitehall in December 2020, including a Christmas party, leaving do, quiz, No 11 flat party and drinks at the Department for Education.

After the newest allegations, of another May 2020 gathering during lockdown, Alistair Carmichael, Lib Dem MP and constitutional reform spokesperson, called for the prime minister himself to be interviewed by Gray.

“The government should not skirt around the issue on this. If they are serious about healing public trust, the head of the No 10 Christmas party inquiry Sue Gray should personally interview the prime minister.”

Angela Rayner, the Labour deputy leader, also called for assurances that Gray would be able to interview the prime minister.

“It is vital that Sue Gray has all the information and access she needs to carry out a full and fair investigation,” she said. “Under the terms of reference of the previous investigation, Simon Case was given assurances he would be able to interview any politicians, officials and party staff he needed to.

“While the terms of reference for the new investigation have not been published, it is paramount that Sue Gray is given the same access – including if she needs to interview the prime minister. No 10 must give such assurances in order to help bolster public faith which has been shaken by this series of revelations.”

Downing Street has given no timeline for the completion of the Gray inquiry, which was taken over by the veteran civil servant last month. Simon Case, the cabinet secretary, had been leading the investigation until it emerged his own private office had held a virtual quiz, attended by some people from their desks.

Johnson is under pressure from Tory backbenchers to shake up his top team over the allegations that parties took place in Downing Street, with speculation that Reynolds could be moved as a result. Some MPs are also pushing for the removal of Dan Rosenfield, Johnson’s chief of staff, with a report in the Sunday Times alleging he contributed to the departure of some women from No 10 after allegedly making them buy sandwiches for his lunch, collect his dry-cleaning and buy presents. Asked whether this was true, a No 10 source said they “totally reject that”.

The continuing furore over the parties is one in a long list of scandals dogging No 10, after it emerged last week that Johnson had not handed over all text messages relating to the refurbishment of his No 11 flat to Lord Geidt, the independent adviser on ministerial interests.

On Sunday, the Lib Dems wrote to Lady Hallett, the recently appointed chair of the official Covid inquiry, asking what steps she is taking to ensure she is not being misled by Johnson in the same way that Geidt was. It also asked for the inquiry to examine the reports of No 10’s lockdown-breaching parties, saying: “The Covid inquiry must look into these allegations and the damage done to public trust in the fight against the pandemic.”

The letter from Carmichael asked Hallett, a retired judge, to set out what steps she is taking to ensure the Covid inquiry has access to all relevant evidence including text and WhatsApp messages, to ensure it has “the confidence of bereaved families and of the public in general throughout the course of its work.”

It also asks if the inquiry will have the power to request that evidence is handed over, and what sanctions if any there will be if key witnesses such as Johnson fail to do so.