The club's membership is a closely guarded secret but is known to include leading figures from the civil service, the law, journalism, publicly funded institutions and the arts.
BBC world affairs editor John Simpson last week tweeted: "Various Garrick Club members including Sting, Mark Knopfler and leading actors and producers have reportedly written to the club chairman saying they'll resign if the membership doesn't vote to accept women next Tuesday."
He added: "Many others like me would also find it impossible to stay."
In March, the head of Britain's spy service Richard Moore resigned after a list of the club's membership was made public for the first time.
Moore sent a message to MI6 employees acknowledging the reputational hit that news of his membership posed to the service -- in particular the risk of it undermining its work to attract more women to join MI6.
Prime Minister Rishi Sunak's most senior policy adviser, the civil service leader Simon Case, also quit his membership.
Founded in 1831 for actors and "men of refinement and education", the Garrick is one of the last such clubs not to allow women in, except as guests of men.
A petition launched in 2021 backing the admission of women attracted the support of Cherie Blair, a leading barrister and wife of former prime minister Tony Blair.
She recalled that in 1976 as a trainee lawyer she was left standing outside while her future husband Tony was allowed in for dinner.
"It's outrageous that so little progress has been made since then," she wrote.
Organisers of the petition said that the large number of judges and senior lawyers who were members deprived women of networking opportunities in a profession in which women were under-represented, particularly in the higher echelons.
A previous vote in 2015 on allowing female members saw the narrowest of majorities -- 50.5 percent -- back continuing the ban on female membership.
The Garrick, located in Covent Garden in central London, offers members overnight accommodation, a restaurant, bars and a library.
Other high-profile figures reported to be members include "Sherlock" star Benedict Cumberbatch and senior minister Michael Gove.
An early member was Charles Dickens.
har/bc
© Agence France-Presse
