Embattled Health Secretary Andrew Lansley faces yet more pressure over his controversial NHS reforms today.
Peers are resuming their scrutiny of the legislation amid speculation that the Cabinet minister's job could be on the line.
The Times attributed the quote to an unnamed "Downing Street source", and even suggested that Labour former health secretary Alan Milburn could be drafted in to take his place.
However, a spokeswoman for the premier insisted he had "full confidence" in Lansley.
"The Prime Minister backs Andrew Lansley and he backs the reforms we are pushing through Parliament in order to deliver a better health service for the future," she said.
"As far as we are concerned, the reforms are going to deliver a better NHS, one which will be freer of bureaucracy and have less political interference."
Cameron is said to have reaffirmed his support for the Health and Social Care Bill at a meeting with Lansley and Liberal Democrat leader Nick Clegg earlier this week.
But the legislation is returning to the Lords less than a week after the Royal College of GPs wrote to Cameron calling for it to be scrapped.
A poll of British Medical Journal (BMJ) readers found that more than 90% believe the reforms should be abandoned.
Labour leader Ed Miliband has also demanded the shake-up should be dumped, warning that opponents have just "three months to save the NHS".
He is due to take on Cameron at Prime Minister's Questions in the Commons later - a forum where he has repeatedly raised the subject.
The Government has already accepted scores of amendments to the Bill, including a guarantee that the health secretary will remain ultimately responsible for providing NHS services in England.
More tension is expected today over the minister's specific duty to provide education and training throughout the service.
There are fears that the key issue of competition in the NHS may not be settled before next month's Liberal Democrat Spring conference.
The scale of opposition at last year's event was partly responsible for the decision to "pause" and rethink the original proposals.
Yesterday two leading lights in the pro-reform NHS Alliance and National Association of Primary Care condemned flaws in the design of the planned changes.
Dr Charles Alessi and Dr Michael Dixon told The Guardian they were concerned that new GP-led clinical commissioning groups will find themselves under the control of a National Commissioning Board, in place of the Primary Care Trusts and Strategic Health Authorities which are being abolished to grant frontline workers more autonomy.
Shadow health secretary Andy Burnham said: "A campaign is clearly under way to scapegoat Andrew Lansley. But it is David Cameron who has put the NHS on a knife edge and it can't afford to have a lame-duck Secretary of State in charge who does not have authority and the personal support of the Prime Minister."
Writing in The Times today, Milburn insisted: "The Health Bill is a patchwork quilt of complexity, compromise and conflation. It is incapable of giving the NHS the clarity and direction it needs.
"It is a roadblock to meaningful reform."
Burnham argued it was the wrong time for the Government to be reforming the NHS, especially when it is facing the "biggest financial challenge in its history".
He told ITV's Daybreak: "This Bill breaks 63 years of NHS history. It puts market forces at the heart of the NHS and will set hospital against hospital in a competitive market. That is not the NHS we have known and trusted for all these years. This was the wrong time to reorganise the NHS."
Burnham said "service change" was needed with more work taken out of hospitals and more care for patients in the community and home.
"The Bill is a distraction from the reform the NHS actually needs," he added.