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.

“The Garrick Club has an absolute right to decide who its members are,” said Simon Jenkins, a columnist at The Guardian and a former editor of The Times of London who is a longtime member. “That said, it is indefensible for any social club these days not to have women as members.”

“Judi Dench, for God’s sake — why shouldn’t she be a member?” he added.

Or Jude Kelly, an award-winning former theater director. Ms. Kelly, who now runs the charity Women of the World, said that excluding women from membership in the Garrick deprived them of access to an elite social circle where professional opportunities inevitably flowed with the brandy.

“We’re in 2024,” Ms. Kelly said. “These are incredibly senior people. Many of them are espousing diversity and inclusion in their professional lives. Being on the inside for a long time makes you complicit.”

The Garrick Club is not the only private club in London that does not admit women: White’s, Boodle’s, the Beefsteak Club and the Savile Club are also men only. But what makes the Garrick unique is its star-studded membership list, which ranges across the worlds of politics, law, arts, theater, and journalism.

Members, based on The Guardian’s leaked list, include the actors Benedict Cumberbatch, Brian Cox and Stephen Fry; Mark Knopfler, the guitarist of the rock band Dire Straits; Paul Smith, the fashion designer; the BBC correspondent John Simpson; Oliver Dowden, Britain’s deputy prime minister; and, yes, King Charles III (on an honorary basis).

The boldfaced names have lent the dispute extra piquancy, especially since many of them would seem the kind of bien-pensant progressives who would abhor any kind of discriminatory policy. Indeed, Mr. Cox, Mr. Fry and Mr. Simpson are among those who have come out publicly in favor of admitting women.

The last time the members voted on the question, in 2015, a slender majority — 50.5 percent — said they supported it. But the club’s bylaws require a two-thirds majority to change the policy on membership, and a new vote, if it were scheduled, would not be held until the summer. A club official declined to comment on the matter.

For all the misgivings that members have about not admitting women, some predict they would still fail to reach the two-thirds threshold. The dispute has, perhaps inevitably, turned bitter, pitting a handful of committed campaigners against a larger, older group, many of whom are fine with women as guests but are reluctant to rock a boat that has sailed grandly since 1831.

In New York City, private clubs like the Union League and the Century Association began admitting women in the 1980s, often under the pressure of legal judgments. But in London, where clubs like the Garrick are more zealous about being social rather than professional networking institutions, defenders argue that the case for preserving male-only membership is more justifiable.

These members say they go to the Garrick to drink wine, unwind and enjoy themselves. They crack jokes they wouldn’t make in mixed company. They are not allowed to conduct business; even pulling papers out of a briefcase is looked down upon.

Some dismissed it as a tempest in a teapot. Jonathan Sumption, a lawyer and former justice in the Supreme Court, said he supported the admission of women, but added that those who opposed it were entitled to their opinion.

“The Garrick Club is not a public body and the whole issue is too unimportant to make a fuss of,” Mr. Sumption said. “It is still a pretty good club.”

Mr. Jenkins, the columnist, agreed, suggesting that some of the news coverage had caricatured the Garrick as a vaguely sinister place where men gather to plot against women. Women, he said, were welcome at the communal table in the dining room, perhaps the club’s most hallowed place.

The only room off limits to women is the members’ lounge, known as Under the Stairs, where men gather after dinner. Yet, as Ms. Kelly and other women note, the most valuable relationships are often formed in such informal settings.

To that extent, the Garrick is different from White’s, an even more exclusive men’s club in St. James’s, where Queen Elizabeth II was the only woman ever invited as a guest. When President Donald J. Trump’s ambassador to Britain, Robert Wood Johnson IV, held lunches there with his senior staff, he could not invite his own political counselor because she was a woman. Female employees at the embassy complained to the State Department, and he was urged to end the practice.

But White’s and its old-line, Conservative-friendly brethren “tend to be high Tory places, where the question wouldn’t arise,” said Alan Rusbridger, a former editor of The Guardian, who resigned from the Garrick more than a decade ago.

“The Garrick membership is more a mix of actors, journalists and lawyers,” he said. “Thus, it’s a more pertinent question.”