In a furious outburst, Mr Cummings accused the Prime Minister of seeking to halt the investigation into the leak of plans for a second lockdown after he was warned a close friend of his fiancee Carrie Symonds could be implicated.
He also claimed Mr Johnson had attempted an "unethical, foolish, possibly illegal" plan to get Tory donors to fund the lavish refurbishment of his official Downing Street flat.
The claims prompted speculation Mr Cummings is bent on revenge after his dramatic departure last year from No 10 amid a bitter internal power struggle with Ms Symonds.
Ministers are braced for further allegations when he appears next month before MPs investigating the Government's response to the coronavirus pandemic.
Mr Johnson sought to brush off the charges saying the public did not "give a monkey's" about such matters.
Asked by LBC radio if he had tried to block the leak inquiry, the Prime Minister replied: "No, of course not."
Downing Street said in a statement that ministers had acted at all times "in accordance with the appropriate codes of conduct and electoral law".
Labour said the Tories were fighting "like rats in a sack" demonstrating a "breath-taking contempt" for the public.
Mr Cummings' onslaught was prompted by briefing to newspapers that Mr Johnson believed he was behind a series of leaks including text messages exchanges in which the Prime Minister promised to "fix" a tax issue for the entrepreneur Sir James Dyson.
In a lengthy blog post he accused Mr Johnson of ordering his aides to make false allegations about him and offered to give evidence to the Cabinet Secretary Simon Case.
He went on to deny a No 10 claim 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" as having been responsible.
The Prime Minister, he said, had been "very upset" as it would cause him "very serious problems" with Ms Symonds if Mr Newman had to be fired and had asked whether the leak inquiry could be stopped.
Mr Cummings said he told Mr Johnson 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".
"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 Cummings 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.
Mr Johnson, who spent Friday campaigning in Hartlepool by-election, insisted he was concentrating on protecting the health of the public, adding: "I don't think people give a monkey's about this issue".
In a statement, Downing Street said that Mr Johnson had "never interfered" in a Government leak inquiry.
In response to Mr Cummings' claims about the funding of the flat refurbishment, the statement said: "At all times, the Government and ministers have acted in accordance with the appropriate codes of conduct and electoral law.
"Cabinet Office officials have been engaged and informed throughout and official advice has been followed."
Earlier a Government statement said that Mr Johnson had met the cost of the flat refurbishment - which reportedly ran to £200,000 - out of his own pocket.
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.
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.