* Women can play the course only if invited by member

AUGUSTA, Ga, April 6 (Reuters) - Political pressure mounted on Friday for Augusta National Golf Club, which hosts the Masters tournament, to bend its ban on women members to allow the female chief executive of tournament sponsor IBM to join the exclusive club.

President Barack Obama and his likely Republican challenger in the race for the White House, Mitt Romney, both said on Thursday the conservative, all-male club should admit women.

On Friday, former presidential candidate and Arizona Senator John McCain joined the chorus, posting on Twitter: "Don't you think it's time Augusta National joined the 21st century - or the 20th - and allowed women members?"

The controversy concerns Virginia Rometty, the 54-year-old former IBM global sales chief tapped in October to head IBM, which is a leading sponsor of the glamorous tournament.

Traditionally, the CEO of a major corporate sponsor is granted club membership and given one of Augusta's famous green jackets, which go to the winner of one of the world's most celebrated sporting events. Four previous male IBM chief executives have been granted membership to Augusta.

At a pre-tournament press conference on Wednesday, Augusta Chairman Billy Payne repeatedly refused to comment on whether Rometty would be offered membership.

"We don't talk about our private deliberations," Payne said in response to a question. "We especially don't talk about it when a named candidate is a part of the question."

The club has stubbornly adhered to the all-male membership rule, even when it came under intense scrutiny over the issue a decade ago.

In 2002, a heated war of words erupted between Augusta's then-chairman William "Hootie" Johnson and Martha Burk, the former chair of the National Council of Women's Organizations.

She attacked the club's males-only policy as sexist. At the time, 15 percent of the club's membership were CEOs, many of them from Fortune 500 companies. Burk argued the exclusion meant women were at a disadvantage at social events where business relationships are often formed.

Johnson said Augusta was just a single-sex organization, like the Boy Scouts or the Girl Scouts.

The controversy led two high-profile members of Augusta to resign from the club. John Snow, former President George W. Bush's nominee for Treasury Secretary, resigned before he was confirmed in that post. Thomas Wyman, a former CBS television network CEO left the club, calling its policy "pigheaded."

IBM has tried to steer clear of the controversy.

"We don't comment on our executives' travel schedules," IBM spokesman Edward Barbini said on Friday in response to a question on whether Rometty would attend the event, which concludes on Sunday.

Augusta's invitation-only membership has been steeped in secrecy since the conservative club opened in 1932. Women are allowed to play the course if invited by a member but cannot become members themselves.

The club does not reveal its full list of members, believed to be around 300, although it is known that some of the most powerful men in industry and finance are members, including Microsoft founder Bill Gates and investor Warren Buffett.

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