On a side street in Covent Garden stands an imposing palazzo-style building, strangely out of place amid the burger joints and neon marquees of London’s theater district. It houses the Garrick Club, one of Britain’s oldest men’s clubs, and on any given weekday, a lunch table in its baronial dining room is one of the hottest tickets in town.

A visitor lucky enough to cadge an invitation from a member might end up in the company of a Supreme Court justice, the master of an Oxford college or the editor of a London newspaper. The odds are that person would be a man. Women are excluded from membership in the Garrick and permitted only as guests, a long-simmering source of tension that has recently erupted into a full-blown furor.

After The Guardian, a London newspaper, put a fresh spotlight on the Garrick’s men-only policy, naming and shaming some of its rarefied members from a leaked membership list, two senior British government officials resigned from the club: Richard Moore, the chief of the Secret Intelligence Service, and Simon Case, the cabinet secretary, who oversees nearly half a million public employees.

Only days earlier, under questioning at a Parliamentary hearing, Mr. Case defended his membership by saying he was trying to reform an “antediluvian” institution from within rather “than chuck rocks from the outside,” a line that provoked derisory laughs. Mr. Moore’s membership seemed at odds with his efforts to bring more racial and gender diversity to the British spy agency, known as MI6.

Now, the club’s 1,300 members are debating the future of the Garrick over lamb chops in the dining room, after-dinner drinks in the lounge under the main staircase and in a WhatsApp group, where they swap fretful messages about the latest developments. Some welcome the pressure to admit women as long overdue; others lament that doing so would forever change the character of the place.