LONDON — An American businesswoman who received thousands of pounds from a government agency that Boris Johnson controlled when he was mayor of London told friends that she and Mr. Johnson were having a sexual affair, according to a British news report this weekend.

The revelation in The Sunday Times of London intensifies the scandal hanging over Mr. Johnson as he tries to navigate an impasse in Parliament over his Brexit plans. Several agencies are investigating accusations that he gave favorable treatment to the businesswoman, Jennifer Arcuri, now 34, by helping her secure sponsorship money and taking her on trade missions that she was not qualified for.

On Friday, a monitor at London’s City Hall referred Mr. Johnson to a police watchdog for a possible investigation into the case, saying that the accusations, if true, could amount to misconduct in public office.

Both Mr. Johnson and Ms. Arcuri have denied any wrongdoing. And a government minister from Mr. Johnson’s Conservative Party dismissed that as “an obviously politicized complaint.” The monitor was a career official who also worked at City Hall during Mr. Johnson’s tenure as London mayor, from 2008 to 2016.