The Sunday Times reported that a furious Boris Johnson suggested the move after the leak of a letter from the Chancellor calling for the easing of travel restrictions ahead of the relaxations announced on Wednesday.
The letter was copied to Grant Shapps's Transport Department.
A senior source told the newspaper: “He said: 'I've been thinking about it. Maybe it's time we looked at Rishi as the next secretary of state for health. He could potentially do a very good job there.' In an open meeting, after ranting about Rishi, he then suggested the Chancellor could be demoted in the next reshuffle."
The newspaper reported that Mr Johnson is not expected to carry out his threat and noted the Prime Minister's reputation for off-the-cuff remarks made "half in jest".
Another report in the Mail on Sunday revealed that Mr Johnson believed Mr Sunak was guilty of 'a failure of political judgement' in writing the letter.
A source said: “The Prime Minister shared Rishi's views. It didn't need to be written. It was designed to be leaked. Boris regarded it as a failure of political judgment. He hadn't even received the letter - and said that maybe Rishi "could do a very good job" at Health instead of the Treasury.”
A Treasury source said: "The Chancellor is solely focused on securing the country's economic recovery and continuing to protect and create jobs."
The leak of the comments, reportedly made in a meeting on Monday, will do little to help relations between No 10 and No 11.
The Chancellor is preparing for a tough spending review later this year as he attempts to repair the public finances following the coronavirus crisis.
That could put him on collision course with a Prime Minister who has promised there can be no return to austerity.
Although a reshuffle is not expected imminently, the Sunday Times reported that Mr Johnson has previously considered International Trade Secretary Liz Truss as a potential chancellor, with Jacob Rees-Mogg as her deputy.
"The PM keeps talking about Liz Truss," a source said. "He's always got on quite well with her. He thinks she's controllable."