Royal Troon Golf Club is consulting members over whether to end its men-only membership policy.
The Ayrshire club, which is hosting this year's Open, shares its facilities with Ladies' Golf Club Troon but both have separate clubhouses.
Troon captain Martin Cheyne said the club had written to its 800 members to "understand their views and feelings on the issue".
Royal Troon and Muirfield are the only two Open hosts to still exclude women.
Last month, golf's governing body the R&A said the issue was "a matter for the club" to decide on.
Troon was formed in 1878 and Cheyne added: "We care very much for the reputation of Royal Troon Golf Club and it is important that the club, much like the wider game, reflects the modern society in which we exist."
Golf will feature at the Olympics in Rio this summer after a 112-year absence and the sport's oldest institutions have taken steps to modernise since its return to the Games was confirmed.