Muirfield Golf Club is looking to hold a fresh ballot over whether to admit women members just six weeks after rejecting the proposal.
The club on the east coast of Scotland announced on May 19 it had failed to reach the two-thirds majority required for change in a vote among its members.
An independently scrutinised postal ballot recorded a vote of 64% in favour of the resolution and 36% against on a turnout of 95% of members eligible to vote.
With the vote so close, the club will now seek authority from its membership at a special general meeting to hold a fresh postal ballot before the end of the year.
Henry Fairweather, captain of the Honourable Company of Edinburgh Golfers - which owns and runs Muirfield - said: "A substantial majority of our members voted for change and many have voiced their disappointment with the ballot result and with subsequent events.
"The Club Committee believes that a clear and decisive vote in favour of admitting women as members is required to enable us to begin the task of restoring the reputation of the club that has been damaged by the earlier ballot outcome."
And in a strongly-worded letter to each member of the club, Fairweather has called on them to 'resolve the situation' after the decision 'brought widespread criticism from around the world'.
He added: "The golfing community has almost unanimously regretted our decision as being bad for the image of the game worldwide. The Press and politicians of all parties have condemned the decision... and there is no doubt the reputation of the Honourable Company has suffered serious damage.
"Considering the best way forward, the committee has concluded the current situation is unacceptable and the club can only suffer more damage if the matter is not resolved as swiftly as possible.
"In conclusion, the committee and I firmly believe it is essential that members should vote in favour of admitting women members. The consequences of not doing so have now been starkly illustrated."
The Honourable Company of Edinburgh Golfers, founded in 1744 and the first to set down the playing rules of the sport, had urged its members in May to vote in favour of the resolution.
Following the result of the ballot the Royal and Ancient Golf Club ruled Muirfield would not be permitted to hold an Open Championship, claiming it would not allow its showpiece event to be "staged at a club which does not admit women as members".