A reshuffle after the May local elections would allow Mr Johnson to promote recently elected Tory MPs pushing for ministerial jobs but could risk creating new ex-Cabinet critics.
Ms Patel has faced speculation that she could be moved from the Home Office amid record numbers of migrants crossing the English Channel in small boats.
But Mr Johnson has backed her proposal for those who arrive in the UK illegally to be sent to have their asylum applications processed in Rwanda.
Asked whether Ms Patel had saved her job by announcing the Rwanda policy, Mr Johnson said: “You’re not going to get any more out of me about politicians or Westminster politics narrowly conceived. We’ve got to focus on the agenda for the country.”
Earlier this month, it emerged that Mr Sunak, along with Mr Johnson, had been fined by police for a breach of Covid rules as part of the “partygate” investigation.
Speaking in Washington for the spring meeting of the IMF on Thursday, Mr Sunak said he was “extremely and sincerely sorry” for the hurt he had caused by attending a birthday gathering for Mr Johnson during lockdown.
“I fully respect the decision that the police have reached,” he told the BBC. “I paid the fixed penalty notice straight away and I am extremely and sincerely sorry for the hurt and the anger that this has caused so many people.”
Asked whether he had considered resigning, he said: “No. I am focused on getting on with the job I have got to do.”