“This goes into a new category — it’s no longer just the tabloid sex scandal thing,” said Sonia Purnell, the author of “Just Boris: A Tale of Blond Ambition.” “Every single ingredient there is alarming. And the fact that he will not answer any questions — or indeed deny any of those allegations — I think that is why some people see blood in the water.”

The scandal came in the midst of a tense week for the prime minister. He is trying to secure a new Brexit deal with European leaders. An early election is approaching. And the Supreme Court is set to rule on Tuesday on whether Mr. Johnson’s decision to suspend, or prorogue, Parliament at the height of the Brexit crisis was legal.

That shutdown was originally intended to keep lawmakers from interfering with his plans to yank Britain out of the European Union by Oct. 31, with or without a deal governing future relations. But it now serves a second purpose in shielding the prime minister from a very public interrogation over the government grants.

“The fact he’s prorogued Parliament means no one can call him to account,” Ms. Purnell said.

As mayor, Mr. Johnson was barred from giving friends improper benefits. He was also bound to declare any private entanglements that could have created conflicts of interest.

Hacker House, a cybersecurity company set up by Ms. Arcuri, did not respond to a message on Monday seeking comment. She told The Sunday Times that the grants “were purely in respect of my role as a legitimate businesswoman.”

The Sunday Times reported that Ms. Arcuri arrived in London for business school after a short career in modeling and films. Her first venture — a series of networking events for entrepreneurs and policymakers in East London — was given a lift by Mr. Johnson, who appeared at four of her events, The Sunday Times reported.

Soon a promotional agency overseen by the mayor began giving the company money. Mr. Johnson also paid regular visits to Ms. Arcuri’s East London apartment, which was outfitted with a dancing pole, sometimes during afternoon work breaks. The apartment building owner told The Sunday Times that Ms. Arcuri described Mr. Johnson as “one of her best friends.”