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Mr Johnson's former top adviser denied he was responsible for the leak of private texts in which he promised to "fix" a tax issue for the entrepreneur Sir James Dyson.
In a lengthy blog posting he also claimed the Prime Minister had tried to stop an inquiry into the leak last year of plans for a second lockdown because it implicated a friend of his fiancee, Carrie Symonds
He said that he had also warned Mr Johnson against plans to have donors secretly pay for refurbishment of his Downing Street flat, saying they were "unethical, foolish (and) possibly illegal".
"It is sad to see the PM and his office fall so far below the standards of competence and integrity the country deserves," he said.
Mr Johnson flatly denied he had sought to block the leak inquiry. Asked about the claim by LBC radio, he replied: "No, of course not."
He added: "I don't think people give a monkey's about this issue.
"What they care about is what were we doing to protect the health of the British public and that's what I care about."
Mr Cummings' attack follows briefings to a number of newspapers, which said Mr Johnson believed he was the source of the leaks about the lockdown and his texts to Sir James as well as stories about the flat refurbishment.
It follows his dramatic departure from No 10 last year amid the fallout from a bitter power struggle with Ms Symonds.
Mr Cummings' onslaught came as Downing Street outlined details of Mr Johnson's communications with Sir James at the start of the pandemic.
The Government also stated formally for the first time that the Prime Minister met the cost of the Downing Street flat refurbishment out of his own pocket.
In his blog post, Mr Cummings accused the new No 10 director of communications Jack Doyle of making false accusations about him "at the PM's request".
He said that while Mr Johnson had forwarded him some of his WhatAapp exchanges with Sir James, they did not include those featured in the leak to the BBC.
He said that he was happy to meet Cabinet Secretary Simon Case to discuss what had happened and for his phone to be searched.
He also denied that he was the so-called "chatty rat" responsible for the leak last October which was seen to have "bounced" the Government into announcing a second lockdown.
He said Mr Case had told both him and Mr Johnson that "all the evidence" pointed to Henry Newman, a special adviser and a close friend of Ms Symonds, and "others in that office".
The Prime Minister had said this would cause him "very serious problems" with his fiancee if Mr Newman had to be fired and asked whether the leak inquiry could be stopped.
Mr Cummings said that he had told him that was "mad" and "totally unethical" and that he could not cancel an inquiry into a leak which had affected millions of people "just because it might implicate his girlfriend's friends".
He said that there now needed to be an urgent parliamentary inquiry into the Government's conduct over the Covid crisis, with the key players required to give evidence on oath.
Speaking on a by-election campaign visit to Hartlepool before Mr Cummings' explosive allegations were published, Mr Johnson refused to be drawn on why his former adviser was suspected in No 10 of leaking against him.
He again insisted that he had been right to talk to Sir James who had been offering to supply ventilators to the NHS amid fears it could run out due to the pandemic.
"I think people aren't so much interested in who is leaking what to whom as the substance of the issue at hand," he told broadcasters.
"The issue is really the question of the ventilators as you will remember James Dyson was offering to make."
Former foreign secretary Lord Hague said he did not know if there was any truth in the claims but said that Downing Street would need to come up with some "good answers".
"If all of these things were true, they would be damaging to anybody, including a prime minister," he told Times Radio.
"But No 10 will no doubt have to give their account of all of these events and I'm sure there will be another side to the argument."
Labour deputy leader Angela Rayner said the Government was "lurching between cover-ups and cock-ups".
"Labour is focusing on jobs, crime and the NHS, while the Conservatives are fighting each other like rats in a sack and slipping deeper and deeper into the mire of sleaze," she said.
"It shows breath-taking contempt for the country."