The former chief aide in No 10 alleged that he had to convince the Prime Minister out of visiting her by warning about the potentially grave consequences.
Downing Street denied the incident described by Mr Cummings in an interview with the BBC took place and Buckingham Palace declined to comment.
Mr Johnson ended up taking a 15-month break from his face-to-face weekly audience with the Queen after meeting her on March 11 2020 and they instead spoke on the phone.
But Mr Cummings, who has been engaged in a war of words with Mr Johnson since leaving No 10 in November, alleged Mr Johnson wanted to visit her a week later, on March 18.
“I said, what are you doing? And he said, I’m going to see the Queen and I said, what on earth are you talking about, of course you can’t go and see the Queen,” Mr Cummings said in an interview.
“He said, ah, that’s what I do every Wednesday, sod this, I’m gonna go and see her.”
But this was five days before Mr Johnson announced the first lockdown on March 23 and he went on to test positive himself for Covid-19 later that month.
“I said to him (Boris Johnson), there’s people in this office who are isolating, you might have coronavirus, I might have coronavirus, you can’t go and see the Queen,” Mr Cummings said.
“What if you go and see her and give the Queen coronavirus? You obviously can’t go.
“I just said if you, if you give her coronavirus and she dies what, what are you gonna, you can’t do that, you can’t risk that, that’s completely insane.
“And he said, he basically just hadn’t thought it through, he said, yeah, holy shit, I can’t go.”
Ultimately the Prime Minister and the Queen switched to holding phone discussions on March 18 as she prepared to socially distance at Windsor Castle.
Their first in-person meeting was not until June 23 this year
Dominic Cummings: The Interview will air on BBC Two at 7pm on Tuesday.