The PM had said no rules were broken but now is one of more than 50 people fined by police. It led opposition politicians to accuse him of misleading Parliament.
Under Government rules, ministers are expected to resign if they knowingly mislead MPs.
Ministerial rules also say they should correct the record at the “earliest opportunity” if they inadvertently tell MPs something incorrect.
MPs return to Westminster on Tuesday, with Mr Johnson pledging to set out his position.
His penalty was for attending a birthday party thrown in his honour in the Cabinet Room in June 2020, when indoor socialising was banned.
After accusations of hypocrisy for not following the rules, and allegations of lying to MPs, Mr Johnson told a press conference in Kent: “You are going to have to wait until I come to Parliament when, of course, I will set the record straight in any way I can.”
The PM has been broadly backed by his party since being fined but dissenting voices are now emerging.
Lord Wolfson quit as a justice minister on Wednesday, saying he had come to the “inevitable conclusion there was repeated rule-breaking, and breaches of the criminal law, in Downing Street”.
Tory ex-Cabinet minister Karen Bradley suggested the PM should go. She said “law-breaking in Downing Street is unforgivable” but the war in Ukraine meant there was a need to “act responsibly so as to not make the situation worse”.
Ms Bradley added: “If I had been a minister found to have broken the laws I passed, I would be tendering my resignation now.”
Defence Committee chairman Tobias Ellwood warned Russian despot Vladimir Putin would exploit Mr Johnson’s position He told Today on BBC Radio 4: “How can a lawmaker also be a lawbreaker? This is not a good look.”