David Cameron has led senior politicians in condemning Muirfield's male-only membership policy as the prestigious Open Championship began at the historic course.
The Prime Minister's spokesman said Mr Cameron had a "great deal of sympathy" with the view that such policies "look more to the past than they do to the future".
Deputy Prime Minister Nick Clegg said the club's membership policy was "anachronistic" and Commons leader Andrew Lansley said it was "reprehensible". Culture, Media and Sport Secretary Maria Miller was boycotting the Open in protest at Muirfield's stance and her Labour shadow Harriet Harman called for a ban on male-only clubs.
The Prime Minister's spokesman said Mr Cameron was not attending the Open and "entirely understands" why Mrs Miller was not going. Asked about all-male clubs, he added: "The Prime Minister has a great deal of sympathy with the view that exclusive memberships of this sort look more to the past than they do to the future."
Mr Clegg addressed the issue on his LBC 97.3 phone-in show, saying: "I was just dismayed and incredibly surprised to hear this still goes on in this day and age. I find it so out of step with everything else that's happening in the rest of society. It just seems so old-fashioned and so anachronistic to just have a golf club saying that just because you are a woman you can't be a member of that club."
Ms Harman, the deputy Labour leader, said: "After the recent successes of female sport it's an embarrassment that this year's British Open, a world class sporting event, is being held at a club which does not admit women members. It's time that Muirfield dragged itself into the 21st century and let women in. It's time to ban men-only sport clubs."
The head of the body that governs women's amateur golf said the issue of single-sex clubs would be addressed "in time". Shona Malcolm, chief executive of the Ladies' Golf Union, accepted having men-only clubs was a "bit of an anachronism in this day and age" but said the sport was in an "evolutionary process". She said any change away from single-sex clubs had to be supported by the sport, and should not be imposed from outside.
She raised the issue the day after Peter Dawson, the chief of golf's governing body, the Royal and Ancient Golf Club, said officials were "very conscious" of views on the issue.
A spokesman for Muirfield said there were "no plans" to change its membership policy. He said: "Muirfield is inextricably linked to The Open Championship and its heritage. This week will mark the 16th time The Open has been staged there. The club welcomes women to play either as visitors or guests year round with full use of the facilities as will be the case throughout the championship."
"We are disappointed that some individuals feel unable to attend this year's Open staged by The R&A at Muirfield. As a Club we conform to the Equality Act 2010 and any change in the membership would be for the members to decide. At this moment there are no plans to change the current membership status."