The prime minister was said have been furious after a letter from Mr Sunak calling for an easing of travel restrictions was leaked to the press last week.
Mr Johnson “ranted” about his chancellor and suggested he should be demoted to health secretary in a meeting with about a dozen officials at Downing Street, according to The Sunday Times.
Allies of Mr Sunak have insisted he remains focused on his job at No 11 amid the row but Labour described the level of dysfunction at the top of government as “disgraceful”.
Bridget Phillipson, shadow chief secretary to the Treasury, said: “The prime minister urgently needs to get a grip on the real challenges facing this country.”
The Labour MP added: “The Covid pandemic continues, tens of thousands of livelihoods are still at risk, the climate crisis threatens our planet but he’s busy picking fights with his own government and threatening to sack the chancellor. It’s completely disgraceful.”
Lib Dem leader Ed Davey told The Independent: “After the calamity of the past two years, the prime minister should look in the mirror and demote himself before worrying about the rest of them.”
Mr Sunak’s letter is said to have been sent to the PM several days before details were published. Whitehall officials blamed the oversight on No 10 staff, who apparently failed to draw the memo to his attention.
In a fit of frustration, Mr Johnson is said to have told his officials at an open meeting: “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.”
Although a reshuffle is not expected imminently, it was reported that Mr Johnson has previously considered international trade secretary Liz Truss as a potential chancellor with Jacob Rees-Mogg as her deputy.
One source said: “The PM keeps talking about Liz Truss. He’s always got on quite well with her. He thinks she’s controllable.”
The latest row over leaked comments, reportedly made in a Downing Street meeting on Monday, will do little to help strained relations between No 10 and No 11.
The chancellor is preparing for a tough spending review as he attempts to repair the public finances following the Covid-19 crisis, which could put him on a collision course with a prime minister who has promised there can be no return to austerity.
The Treasury’s review into the costs in achieving the target of “net zero” carbon emissions by 2050 has reportedly been delayed because of concerns that the government’s plans could hit taxpayers and cost the Tory Party votes.
Tory MP Craig Mackinlay, chair of a new Net Zero Scrutiny Group of backbenchers, told The Sunday Telegraph, the government’s ambition to subsidise green technologies could be seen as following Labour policy under Jeremy Corbyn.
He said: “To ape the failed policies of an extreme Labour politician does not seem to be the way of electoral success. I’m very pleased the Treasury is actually thinking of this with a financial head on rather than just a warm feeling.”
Labour urged the government to step up its game on climate crisis, accusing Mr Sunak of “dragging his feet” on a plan to pay for carbon-cutting measures.
Luke Pollard MP, shadow environment secretary, said: “We are in the decisive decade in our fight to avert catastrophic climate breakdown and we desperately need serious leadership.
“Instead, we have a prime minister who is more interested in vacuous boosterism than the hard yards of policy work … and a chancellor who is dragging his feet on publishing the crucial net-zero review.
“The government must urgently step up its game or risk failure domestically, to meet our targets, and internationally, as hosts of Cop26.”
Following reports Mr Johnson threatened to demote Mr Sunak, a Treasury source said: “The chancellor is solely focused on securing the country’s economic recovery and continuing to protect and create jobs.”
It comes as the government was condemned for refusing to rule out new licences for oil and gas in the North Sea or a new coal mine in Cumbria.
Climate minister and Cop26 summit president Alok Sharma refused to criticise plans for further fossil fuel extraction, telling The Observer: “Future [fossil fuel] licences are going to have to adhere to the fact we have committed to go to net zero by 2050 in legislation.”
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