The Prime Minister insisted on Sunday his own healthcare is “not really relevant” as he declined to set out whether he relies on the public health system.
Labour said Mr Sunak gave the impression of being a leader who “not only doesn’t use the NHS but doesn’t understand the scale of the challenges”.
Mr Sunak has made reducing the NHS waiting lists one of his key priorities over the next two years and has held emergency talks with health leaders to alleviate the crisis.
But unlike Conservative former prime minister Margaret Thatcher, he has refused to say whether he has used private healthcare to avoid the queues himself.
Under sustained questioning, Mr Sunak told BBC One’s Sunday With Laura Kuenssberg programme: “As a general policy I wouldn’t ever talk about me or my family’s healthcare situation.
“But it’s not really relevant, what’s relevant is the difference I can make to the country.”
He said healthcare is a “private” matter, adding that discussing his own situation is “a distraction from what the real issue is, and the real issue is are we making sure there’s high quality healthcare for the country”.
“But when it comes to the private sector in general, we should be making use of the independent sector. I don’t have any problem with that whatsoever,” he added.
In November, The Guardian newspaper reported that Mr Sunak is registered with a private GP practice in west London that guarantees patients with urgent concerns will be seen “on the day”.
Royal College of Nursing general secretary Pat Cullen, who has been leading strikes in an attempt to secure nurses a better pay deal, urged the Prime Minister to “come clean”.
“I think as a public servant, you ought to be clear with the public whether or not you are using private health cover,” she told Kuenssberg.
“That’s about being open, it’s about being transparent, and it’s about honesty.
“I think he needed to come clean. As a public servant he is elected by the public, so he is accountable to the public, and when you’re accountable to the public, you have to be honest with them.”
Shadow health secretary Wes Streeting said he does not use private healthcare as he sought to paint Mr Sunak as being out of touch.
“I thought the Prime Minister in that interview gave the impression of someone who not only doesn’t use the NHS but doesn’t understand the scale of the challenges or have a plan to deal with the fundamental problems,” the Labour MP told the BBC.
“Because, yes, you can get people around the table in Number 10 for a photo op, yes you can do more sticking plasters to get through this winter…
“But we need fundamental change in the NHS to deal with what is the biggest crisis in its history, and that’s what Labour is looking to do.”