The leader of the opposition Labour Party, Sir Keir Starmer, has called Mr Johnson a “coward” for refusing to apologise to Parliament.
On Wednesday, 17 November, Starmer claimed Mr Johnson had "led his troops through the sewers and he can't even say sorry."
He added: "Beneath the bluster he still thinks it is one rule for the public and another for him and his mates."
The Speaker, Sir Lindsay Hoyle, called on Starmer to withdraw the remark and said the word coward was "not what is used in his House."
"I withdraw it, but he's no leader," Keir Starmer replied, although he went on to repeat the insult on Twitter.
Mr Johnson accused Starmer, a barrister, of “refusing to explain” his work for the law firm Mishcon de Reya and joked: “His mischconduct is absolutely clear."
The prime minister has told the Speaker of Parliament, Sir Lindsay Hoyle, he will support plans to ban MPs from taking second jobs as paid consultants and advisers.
Service to honour British MP David Amess in London
© REUTERS / POOL / Service to honour British MP David Amess in London
"It is imperative that we put beyond doubt the reputation of the House of Commons by ensuring the rules which apply to MPs are up to date, effective and appropriately rigorous," Mr Johnson said in the letter.
But he has not apologised.
On Tuesday, 16 November, the Leader of the House of Commons, Jacob Rees-Mogg, accepted he was partly to blame for the government’s original decision to overlook Owen Paterson’s actions.
Mr Rees-Mogg told the Commons on Tuesday: “The tragedy that afflicted Mr Paterson coloured and clouded our judgement, and my judgement, incorrectly and it is as simple and as sad as that ... I regret that the amendment conflated an individual case with more general concerns, that was a mistake."
Parliament finally passed a motion reversing the government’s original decision on 3 November which rejected the third report of the committee on standards.
Paterson - whose wife committed suicide last year during a parliamentary probe into his conduct - resigned earlier this month and triggered a by-election in his North Shropshire constituency which will take place next month.