Johnson has so far weathered calls to quit over parties held in his Downing Street residence and office during coronavirus lockdowns.
"After a great deal of soul-searching, I have reached the conclusion that the Prime Minister should resign," Peter Aldous, the Member of Parliament for Waveney, eastern England, tweeted.
"I have never taken such action before and had hoped that I would not be put in such an invidious position. Whilst I am conscious that others will disagree with me, I believe that this is in the best interests of the country, the Government and the Conservative Party."
In order to trigger a leadership challenge, 54 of the 359 Conservative members of parliament must submit letters of no confidence.Â
The number of letters currently submitted is not known but several have publicly said they have sent theirs.
A limited report by senior civil servant Sue Gray on Monday concluded that alcohol-fuelled events had taken place at 10 Downing Street when rules banning all but limited social mixing were in force, and there had been "serious failures" of leadership.
Meanwhile, Deputy Prime Minister Dominic Raab defended Johnson, saying he "believes he acted in good faith at all times" and suggesting the prime minister does not think he personally did anything wrong.
Asked whether Johnson should quit if he is issued with a fixed penalty notice, Raab said: "Let's wait and see... Allow the police to conduct their investigation and see, when they have ascertained the facts, quite what they conclude."
Police are examining hundreds of documents and photographs in relation to the 12 events in 2020 and 2021 held while England was under coronavirus restrictions.
Despite Aldous' actions and the lingering concerns of other MPs, Johnson's position appears to be secure for now following his promise to make major changes to his Downing Street operation.
Johnson also committed to publishing a fuller version of Gray's report once the police investigation has concluded - although it is not clear how detailed that will be and whether it will include the evidence submitted to the police.
"It's not clear to me that there is anything more, other than any conclusions that she will draw once that investigation is concluded, that will come forward," Raab told LBC.
Johnson has also highlighted a greater role for Australian election guru Sir Lynton Crosby in an effort to bolster his support on the Conservative benches.
Raab said Sir Lynton "has got a good strategic nose and a good sense of the direction of public opinion".
Labour leader Keir Starmer repeated his call for Johnson to quit.
"It's no good him trying to blame politicians in general. There's one person at the centre of this who has caused all of those problems and is subject to a criminal investigation because of his own behaviour," Keir told the BBC.
with reporting from PA
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