Downing Street parties on 27 November and 18 December, and at the education department on 10 December will be examined for Covid rule breaches.
Paymaster General Michael Ellis said any potential criminality uncovered would be reported to the police.
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Labour called for the prime minister to resign if he is found to have misled MPs about the parties.
Shadow Cabinet Office minister Fleur Anderson questioned whether the investigation was "serious" and whether more parties needed to be included in it.
Boris Johnson announced on Wednesday that an inquiry would be carried out by top civil servant Simon Case, after a video from last year emerged of government staff joking about a party that took place on 18 December - a day on which more than 400 Covid deaths were reported.
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Government spokeswoman Allegra Stratton resigned later on Wednesday over the video, obtained by ITV News, in which she appeared.
And Labour has called for Commons Leader Jacob Rees-Mogg to apologise after joking at an event on Monday that those attending would not be "investigated by the police in a year's time".
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The Metropolitan Police has ruled out an investigation for the time being into the 18 December Downing Street gathering, citing an "absence of evidence" of any criminal wrongdoing.
On Wednesday, Mr Johnson said he had been "repeatedly assured" that "there was no party and that no Covid rules were broken" on that date - at which time London was under Covid rules banning people from meeting indoors.
But he said he had asked Mr Case to "establish all the facts" and that if rules had been broken those responsible would face disciplinary action.
The 10 December gathering took place during the same period of restrictions, while on 27 November England was in full lockdown.
Paymaster General Michael Ellis says police will be notified if the law was broken
Mr Ellis told the House of Commons on Thursday: "The primary purpose of the cabinet secretary's investigation will be to establish swiftly a general understanding of the nature of the gatherings, including attendance, the setting and the purpose, and with reference to adherence to the guidance in place at the time."
He added: "As with all internal investigations if, during the course of the work any evidence emerges of behaviour that is potentially a criminal offence, the matter will be referred to the police and the Cabinet Office's work may be paused."
Mr Ellis also said: "All ministers, special advisers and civil servants will be expected to cooperate with this investigation."
But Mr Case will not look into reports of social gatherings in Downing Street on 13 November or a party at Conservative Party headquarters on 14 December, over which four members of staff have been disciplined.
'Just be straight'
The prime minister has repeatedly said no Covid rules were broken in Downing Street, but Ms Anderson asked: "If this investigation finds out that the prime minister has misled the House, will he resign?"
She also asked: "Are there more parties that we need to hear about?"
"People across the country are angry," Ms Anderson said, adding: "Will the government just be straight with the British people?"
The Liberal Democrats have written to Mr Case asking him to investigate any event "where Covid laws banning gatherings may have been breached" in government buildings.
The party's justice spokesperson Wera Hobhouse also called on Mr Case to demand to see WhatsApp messages sent by the prime minister at the time of any gatherings.
"The public must see justice is done on this, and if that means Boris handing over his phone, then so be it," she added.
Watch Allegra Stratton make a tearful apology outside her home: "I will regret those remarks for the rest of my days"
Several Conservative MPs were also critical of the government.
One of them, Bob Blackman, told the Commons: "One of the key issues here is those that are making draconian rules not only have to live by the letter of the rules but by the spirit of the rules as well."
Another, Philip Hollobone, said: "My constituents are very angry indeed about reports of Christmas parties in Downing Street during what was a very large second wave of Covid, and the behaviour was totally inappropriate and possibly criminal."
A number of other events are known or reported to have taken place during Covid restrictions last year, including the three being investigated by Mr Case.
Here is what we know so far:
13 November 2020: Sources have told the BBC there were impromptu drinks to mark the exit of the former director of communications at No 10, Lee Cain - but it was staff having drinks at their desks and was over by 20:30.
13 November 2020: Also the date Dominic Cummings left as chief adviser to the PM. Sources told the BBC that several Downing Street staff members attended a gathering with Carrie Johnson in the flat where the prime minister and his wife live above No 11. A source said music was blaring and could be heard elsewhere in the building. But others said to be at the party denied it took place, as did a spokeswoman for Mrs Johnson.
27 November 2020: Another No 10 aide, Cleo Watson, left Downing Street marked by an event. Again, sources said it was not formally organised, but people were having drinks and Mr Johnson made a speech. Mr Cummings has tweeted that no party took place on this date.
10 December 2020: The Department for Education has confirmed it had a gathering in the office to thank staff for their work during the pandemic. Drinks and snacks were brought by those who attended and no outside guests or support staff were invited.
14 December 2020: The Conservatives have confirmed reports that there was an "unauthorised social gathering" in the basement of their party's building, held by the team of the London mayoral candidate at the time, Shaun Bailey. It was described as "raucous". Four Tory staff members have been disciplined.
18 December 2020: The now infamous Downing Street Christmas party took place on this date, as first reported by the Mirror. A source told the BBC there was food, drinks and games at the gathering that went on past midnight.
Another December party: We are still trying to pin down the date of this one, but multiple sources have told the BBC there was a Christmas quiz for No 10 staff in the Cabinet Office, with invitations sent out in advance via email, telling people to form teams of six. Downing Street says it was a "virtual gathering" but sources at the event say there were groups sat together in the room.
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