Boris Johnson has signed a letter calling for David Cameron to stay on as prime minister whatever happens in next week's EU referendum.

The former Mayor of London has called on other Brexit-backing MPs to put their name to the letter, currently circulating Westminster, which says that even if the UK votes to leave, Cameron should keep the keys to Number 10, according to a report in The Sun.

In a move which could sow the seeds for a post-referendum healing of the party – and ensure he is still in the running for a big cabinet post in the reshuffle that will surely follow – Johnson wants to ensure a period of stability follows 23 June.

Johnson came under intense pressure from Remain campaigners in a televised debate last week facing accusations he had opportunistically backed the Leave side just to try to oust Cameron.

Read more: What time will we know the EU referendum result?

Michael Gove, head of the Vote Leave campaign group and personal friend of the prime minister, has also signed the letter.

It just days after more than 60 Conservative MPs threatened to throw their own government into chaos, potentially triggering a leadership contest, if George Osborne tries to put forward a tax-rising emergency post-Brexit budget. Senior government officials on the Leave side, including Johsnon, Gove and Priti Patel, all refused to lend their name to that letter, however.

Tory figures were previously concerned that whatever the vote in the referendum there could be enough discontent in the parliamentary party to bring Cameron's position under question. If 50 sitting MPs write to the chair of the 1922 Committee – the backbench MPs group – then it automatically triggers a leadership contest.