International Business Machines Corp. (NYSE: IBM), the 101-year-old computer company whose first woman CEO started on Jan. 1, declined to comment Thursday about her possible appearance at the all-male Augusta National Golf Club.

Historically, the chief executive of a corporate sponsor of the Masters tournament, which began Thursday at Augusta, personally congratulates the winner at a ceremony. But the Augusta, Ga., club that hosts the Masters has been men-only since its founding in 1933.

If tradition holds, IBM CEO Virginia Rometty, 54, would be barred from participation. Another tradition is that the chief of a sponsor is elected a member of Augusta National, as were Rometty's three immediate predecessors including Samuel Palmisano, 60, who is now IBM's chairman.

Two other companies are Masters sponsors -- oil giant Exxon Mobil Corp. (NYSE: XOM) and wireless carrier AT&T Inc. (NYSE: T). Their male CEOs -- Rex Tillerson, 60,  and Randall Stephenson, 51, respectively -- have been elected to membership.

At a news conference Wednesday, Augusta National chairman Billy Payne declined to respond directly to questions about Rometty's eligibility or potential appearance at the club. He referred to her as a named candidate, the New York Times reported.

Later, Payne said Augusta has appointed a very smart and motivated team of members who have been given the charge of determining what more we can do.

Augusta National admitted its first black member, Ron Townsend, in 1990.

President Barack Obama thinks women should be admitted at Augusta, White House spokesman Jay Carney said Thursday. It's long past the time when women should be excluded from anything. 

At IBM, Rometty is one of three women directors on a board of 14, along with diplomat Joan Spero and Shirley Ann Jackson, president of Rensselaer Polytechnic Institute. Jackson and fellow board member Kenneth Chenault, chairman and CEO of American Express Co. (NYSE: AXP), are black.

IBM also has several women in top positions, including corporate secretary Michelle Browdy; Linda Sanford, vice president for enterprise transformation; and Bridget van Kralingen, senior vice president for global business services.

The Armonk, N.Y., company employs 433,000 people but doesn't report a gender breakdown.