Figures close to Mr Sunak have sought to play down the impact of the comments - said to have been made by Boris Johnson in a meeting last Monday - as tensions between Number 10 and Number 11 rise.
It was reported over the weekend that the Prime Minister furiously threatened to demote the Chancellor to Health Secretary, after the leak of a letter in which Mr Sunak called on the Prime Minister to relax coronavirus travel restrictions.
With a spending review due in the autumn, there have been suggestions that North Yorkshire Richmond MP Mr Sunak is looking to tighten the purse strings following 18 months of pandemic spending.
However several of the Prime Minister’s big election pledges which gave him his overwhelming Parliamentary majority in 2019 - such as levelling up and a promise to reach net zero carbon emissions by 2050 - are thought to need further cash injections to stay on track.
On Sunday, reports emerged that Boris Johnson had threatened to demote the Chancellor - and Richmond MP - Mr Sunak to the position of Health Secretary following the leak of a letter written by Mr Sunak last week, urging the Prime Minister to relax travel restrictions.
A senior source told the Sunday Times that Mr Johnson “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.’
The source went on: “In an open meeting, after ranting about Rishi, he then suggested the Chancellor could be demoted in the next reshuffle.”
A Treasury source said: “The Chancellor is solely focused on securing the country’s economic recovery and continuing to protect and create jobs.”
The Sunday Times 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”.
Mr Sunak has been touted by many as the possible next Prime Minister, having received high approval ratings from the public throughout the pandemic.
Labour have told the Prime Minister to “get a grip” and stop “picking fights”, calling the reports “completely disgraceful. .
Labour’s shadow chief secretary to the Treasury, Bridget Phillipson, said :“The Prime Minister urgently needs to get a grip on the real challenges facing this country.
She went on: “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.”