Mr Johnson set a target of millions of tests per day, with results delivered in as little as 20 minutes, in “the near future”.

Other medical experts have suggested the target is unrealistic, while a report in the British Medical Journal says leaked memos show the cost of the programme could be as high as £100bn.

Sir John said the Government should “back off” the idea of ten million daily tests, adding: “It’ll be two or three million I think, in the first instance.”

“The real question is whether they can get supply in a position where it can outpace demand, and that’s the challenge at the moment.”

The Government says it is prioritising areas with the highest infection rates, despite reports that there are no available tests in the UK’s ten worst-affected regions.

Mr Hancock has said the problems obtaining tests stem from a sharp rise in the number of people coming forward for tests who are not eligible, because they do not have symptoms.

It is thought that the return to school has placed pressure on the system because parents are requesting tests for their children if another student in their class has symptoms.

The Prime Minister’s latest testing target follows the 100,000 testing target set by Matt Hancock to be achieved by May 1.

The Health Secretary then promised 250,000 tests per day “very soon”.

The current target is 500,000 a day by the end of October.

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