Mr Cummings, who quit as the Prime Minister’s senior adviser last year following a behind-the-scenes power struggle in No 10, published the statement after an internal inquiry was launched into how messages between the Prime Minister and billionaire Sir James Dyson were leaked to journalists, amid an ongoing row over lobbying in Whitehall.
Multiple reports said Downing Street sources were pointing the finger at Mr Cummings.
“Dominic is engaged in systematic leaking,” a source told The Times. “We are disappointed about that. We are concerned about messages from private WhatsApp groups which have very limited circulation.”
But Mr Cummings said today, in a lengthy blog post, that while he was not the source of the link he did have other messages between Sir James and the PM.
And in a series of other damaging allegations he said: “It is sad to see the PM and his office fall so far below the standards of competence and integrity the country deserves.”
Mr Cummings said he had also been cleared of leaking information ahead of November's lockdown in the so-called "chatty rat inquiry".
He said he had told the Prime Minister it would be “mad and totally unethical” to stop that inquiry when it began to suspect a friend of his fiancée, Carrie Symonds.
He said Cabinet Secretary Simon 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.
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 also said he had told Mr Johnson that plans over how to fund the renovations to his flat were “unethical, foolish, possibly illegal and almost certainly broke the rules on proper disclosure of political donations”, but he had not leaked information about that either.
The Government also stated formally for the first time today that the Prime Minister met the cost of the Downing Street flat refurbishment out of his own pocket.
A No 10 spokesman declined to deny suggestions the controversial former aide was the suspect when repeatedly questioned during a Westminster briefing with journalists.
“I am not going to comment on speculation and, of course, it is for the Cabinet Office to take that work forward, I am not going to pre-empt it,” he said.
Downing Street today published details of Mr Johnson’s communications with Sir James and the PM said: “Let me tell you, if you think that there’s anything remotely dodgy, or rum, or weird or sleazy about trying to secure more ventilators at a time of a national pandemic and doing everything in your power to do that then I think you’re out of your mind.”
Asked if he needs to rethink how he communicates with people, Mr Johnson said: “No. I think I need to maintain… to be in touch with people.”
Mr Cummings said he has “made the offer to hand over some private text messages, even though I am under no legal obligation to do so, because of the seriousness of the claims being made officially by No10 today, particularly the covid leak that caused serious harm to millions”.
However, he added that this “does not mean that I will answer every allegation made by No10”.
He said the “proper way for such issues to be handled” would be through a Parliamentary inquiry into the Government’s conduct over the Covid crisis.
He said this “ought to take evidence from all key players under oath and have access to documents”.
He added: “Issues concerning covid and/or the PM’s conduct should not be handled as No10 has handled them over the past 24 hours.
“I will cooperate fully with any such inquiry and am happy to give evidence under oath.
“I am happy for No10 to publish every email I received and sent July 2019-November 2020 (with no exceptions other than, obviously, some national security / intelligence issues).”