The club still has the right to make its own decision, but "[Obama's] personal opinion is women should be admitted to the club," spokesman Jay Carney said during the Thursday White House briefing.
When later asked if Obama would play at a men-only club, Carney said the president had not specifically addressed that scenario during their conversation on the subject.
The White House's statements come amid a coordinated effort by the Democratic Party and the Obama campaign to label the Republican Party as anti-women.
Carney was also asked to explain Thursday why the White House is choosing to host the Forum on Women and the Economy Friday. But he offered no direct explanation for the focus, and responded by saying that other sectors of the population will be the centerpiece of future events.
A Gallup poll of registered voters in battleground states released Monday shows an increasing gender gap between Obama and Romney as Democrats have begun courting the female vote ahead of Election Day.
Democrats are arguing that Republicans are alienating female voters this cycle by focusing on women's issues such as contraception, abortion and federal funding for women's health care provider Planned Parenthood.
Republican National Committee Chairman Reince Priebus on Wednesday dismissed the suggestion of a "GOP war on women," saying on Bloomberg Television's "Political Capital With Al Hunt": "If the Democrats said we had a war on caterpillars and every mainstream media outlet talked about the fact that Republicans have a war on caterpillars, then we'd have problems with caterpillars. It's a fiction."
Democrats immediately seized on those comments, citing them as evidence the GOP is out of touch.
Democratic National Committee Chair Debbie Wasserman Schultz issued the following statement Thursday amid a blast of Democratic emails highlighting Priebus' comments:
To have the head of the GOP say these attacks on women are as fictional as a 'war on caterpillars' is callous and dismissive of what matters to women and completely out of touch. Chairman Priebus and the Republican Party know they have a serious problem on their hands: Mitt Romney and the Republican Party are seeing a serious deficit among women voters, who are simply fed up with Romney and the other Republican candidates advocating for policies that would hurt women and take us backward instead of focusing on jobs and restoring economic security for the middle class.
"From the Dept. Of Clueless, RNC chair says GOP problems with women are contrived by media," Obama's former senior adviser David Axelrod tweeted.
"Women are already abandoning the Republican Party in droves because of their antiquated positions on women's health and out-of-touch policies on the middle class," Obama for America deputy campaign manager Stephanie Cutter said in a statement Thursday. "Reince Priebus' comments today only reinforce why women simply cannot trust Mitt Romney or other leading Republicans to stand up for them."
Priebus said Wednesday that Republicans this election will be making their case "to women and everyone in this country."
"We can do better in this country in regards to jobs and the economy," Priebus said.
Update 3:57 p.m. ET: Mitt Romney was asked later Thursday to weigh in on Augusta National's men-only policy. His response, via Emily Friedman of ABC News:"Well of course. I'm not a member of Augusta...I don't know I would qualify-- my golf game is not that good but certainly if I were a member and if I could run Augusta which isn't likely to happen but of course I'd have women in Augusta, sure."
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