Muirfield's members have been urged not to "make the same mistake twice" after the club announced plans to seek a second vote on whether to admit women members.
A two-thirds majority was required to alter the Scottish club's constitution but only 64% voted in favour and 36% against in May.
Of the 648 members who were eligible to vote, 616 took part with 397 voting in favour and 219 against. A total of 411 yes votes was required.
The R&A responded immediately by removing Muirfield from the Open rota, but chief executive Martin Slumbers said at the time they would reconsider if "the policy at the club should change" and responding to the news of a potential second ballot, an R&A spokesman said on Monday: "We welcome this development."
In a statement released to Press Association Sport, leading charity Women in Sport said: "Last month's decision by members of Muirfield to vote against allowing women golfers to join was a dreadful step backwards for both the club and for the sport in general.
"It is hardly surprising Muirfield's reputation has been significantly damaged, as it is clear they currently fail to offer an environment that is welcoming to all.
"While we hope a re-vote will see their membership policies brought into line with gender equality expectations of the 21st century, ensuring women receive the full benefits of this sport should have been happening far sooner.
"We hope the members of Muirfield, who have been afforded a second chance on this issue, will not make the same mistake twice."
The Honourable Company of Edinburgh Golfers (HCEG), which owns and runs Muirfield, has called a special general meeting to seek authority from members to hold a new ballot before the end of the year.
"A substantial majority of our members voted for change and many have voiced their disappointment with the ballot result and with subsequent events," HCEG captain Henry Fairweather said.
"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."
Royal Troon, which hosts the Open next month, is the only other club on the rota which does not currently admit women, but brought forward a vote on the issue to 1 July and said that "opening the club to women members is supported by over three quarters of the members who participated in the club's survey."
