Muirfield plans to hold another vote on female membership, just weeks after members rejected proposals to allow women golfers to join the club.
The decision taken in May was greeted with anger in and out of the game and the course was stripped of the right to host golf's Open Championship.
The East Lothian club held a ballot at the end of a two-year consultation on membership but failed to get the two-thirds majority of its 648 eligible voters required to change policy.
The Honourable Company of Edinburgh Golfers, which runs Muirfield, said it is now "seeking a fresh ballot of its membership on admitting women as members of the Club".
Of the 616 members who voted last month, 397 - or 64%, voted in favour of admitting women, while 219 - or 36%, voted against - falling narrowly short of the required two-thirds majority for change.
The club's committee supported allowing female members and will propose another postal ballot before the end of 2016 at a Special General Meeting of members.
Approving female members will begin "restoring the reputation of the club", captain Henry Fairweather believes.
He 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."
Some Muirfield members reportedly had concerns over ''slow play'' and future changes to club rules if women joined.
Politicians and leading golfers blasted the decision in May and urged another ballot.
The Honourable Company of Edinburgh Golfers was founded in 1744 and is said to be the oldest recorded golf club in the world.
It set down the original rules of golf in 1744 and has hosted The Open 16 times.
The course last hosted the prestigious tournament in 2013, but will be unable to do so again until it changes membership rules.
Troon is now the only remaining club on the Open rota to have a male-only membership policy, but the South Ayrshire club's committee wants members to approve women joining before it hosts the Open in July.
Troon has considered itself a ''special case'' as it shares facilities with the Ladies Golf Club, Troon, and both clubs will shoulder the responsibility of hosting this year's Open.