Muirfield's decision to hold a second ballot on admitting female members "delighted" the head of the R&A.
Chief executive Martin Slumbers removed the Scottish course from the Open rota when a vote to change its all-male membership policy failed to gain the required two-thirds majority last year.
Slumbers told BBC Sport that Muirfield's "closeness with the Claret Jug is very important".
The result of the latest ballot is expected to be published next month.
In a wide ranging-interview, Slumbers also:
- Backed calls for the Tokyo club scheduled to stage the next Olympic golf tournament to have unrestricted membership for women.
- Insisted President Trump's Turnberry remains on the list of Open venues.
- Admitted the post-Brexit crash in the value of the pound is exerting pressure on the Championship's prize fund.
- Called on all golfers to make sure they shout warning cries of "fore" if there is a danger of their ball striking someone else on the course.
'We need more positive views of this wonderful game'
Slumbers said Muirfield not overturning its all-male membership policy last year had been damaging for the game.
"I think it created a lot of negative press," he said. "We need more positive views of this wonderful game and growth that is going on in the game.
"I'd like to see far more of those stories, plus I do believe that if we are going to grow participation in the game, family golf is at the heart of that strategy."
Muirfield will only return to the Open rota if it changes its membership policy.
'Olympic charter extremely clear on equality'
International Olympic Committee vice-president John Coates has told Reuters that women must have full membership rights at the host club for the Tokyo Games in 2020.
The private Kasumigaseki Country Club, scheduled to stage the men's and women's tournaments, forbids females from playing on Sundays and does not allow them full membership.
"The IOC's charter is extremely clear on equality," Slumbers said.
The R&A is part of the International Golf Federation, which runs the sport in the Olympics, and Slumbers stressed the IGF fully supports this policy.
"I think there would be serious reconsideration on the venue if there wasn't a change," he added.
Trump and Turnberry - 'We are in uncharted territory'
Donald Trump's presidency of the United States is having no effect on his Turnberry course's chances of staging another Open.
"Turnberry remains absolutely as one of our nine golf courses," Slumbers said. "I also said last year that it's clear 2020 and '21 did not involve Turnberry in that discussion, and we will be thinking about '22 not for at least another year."
Slumbers is keen to separate the sport from politics but suggested having the president of the United States so prominent in golf has changed the landscape.
"For all of us in the game, we are in uncharted territory here with the president's family owning golf courses," he added.
"We're all learning as we go. But I think it's important for us that we understand where the game is and make sure we keep to that, without ignoring all the other factors that go around it."
And it seems, like Rory McIlroy, the R&A boss would accept an invitation to play golf with the president.
"With all senior people in the world, I think it's polite and respectful to listen to them and work with them," Slumbers said.
"It's very important that we work with the president if Turnberry did come back on. It would be foolhardy not to."
Dollars for pounds?
Elsewhere, the effects of another political change are being felt at the R&A's St Andrews headquarters.
It is considering paying Open prize money in dollars following the post-Brexit collapse in the value of sterling.
Pressure is growing on the Championship purse at a time when the US Open is offering a record prize fund of $12m (£9.6m).
Last year's Open was worth £6.5m, which at current rates is worth a fraction over $8m.
"It's certainly an issue," Slumbers admitted. "When dollar/sterling moves 1.50 to 1.25 that has a serious impact for us."
The prize money for this year's Championship at Royal Birkdale will be announced in June.
"I'd hate to see the Open not have prize money at the top end," Slumbers said.
"This is a professional game, the players play for money so [paying out in] dollars is one option that we are considering quite carefully."
'They should be shouting 'fore' more often'
American player Pat Perez has been heavily criticised for not shouting "fore" before his ball struck three fans in separate incidents during the third round of last week's Genesis Open.
Ireland's Shane Lowry tweeted: "What's it gonna take for players to start shouting fore? A signed ball or glove is no good to anyone if they are seriously injured."
Slumbers stressed it is R&A policy to remind players of their responsibilities in this regard before rounds in the Open.
"Putting spectators at potential risk is something that can happen in professional golf when the fairways are lined," he said. "And so, for the players and caddies, it's only right that they should be shouting 'fore' more often.
"I think there are enough people who have the same view as I do across all the tours, who think that the etiquette is extremely important to golf as a game and a product."