Writing on Twitter on Monday, Burley said she had been at a "Covid compliant" restaurant on Saturday and had later "popped into another" venue to use the toilet.
Amol Rajan said she was one of a party of 10 people at the Century Club, a private members' club on London's Shaftesbury Avenue. Her group took up two tables, with six people on one and four on the other.
Burley then went onto Folie restaurant, where she used the toilet, before moving on to a private residence where individuals from at least three households mixed, Rajan said.
"This is a source of deep anxiety among Sky News bosses," Rajan said. "There is fury within the Sky News newsroom at the compromising of the brand, especially after Sky News has had a strong year."
Rigby, Rashid and Washington are reported to have been present during the evening, although it's not known which parts they attended.