An openly gay nominee to the vacancy on the U.S. Supreme Court might be acceptable, Sen. John Cornyn (R-Texas) said Monday.

Cornyn, a member of the Senate Judiciary Committee, called on President Barack Obama to make a measured choice in filling the pending Supreme Court vacancy.

Cornyn, a former Texas Supreme Court justice, was initially reluctant when asked if he would support an openly gay nominee — as Sen. Jeff Sessions (R-Ala.), the Judiciary Committee's ranking Republican, said last year he might.

"I'd have to think about that," he said. "As long as it doesn't interfere with their job, it's not a particular issue."

Otherwise, Cornyn said Obama's nominee to replace Justice John Paul Stevens should be "somebody who is obviously qualified by virtue of their experience and with the right kind of judicial temperament."

"By that I mean somebody who is satisfied with being an umpire and doesn't want to be a player," Cornyn said, referring to a common criticism that some Senate Republicans used last year against Justice Sonia Sotomayor during her confirmation process."

Asked if he had any litmus tests for the nominee such as abortion rights or gun rights, Cornyn said no and that he wanted to wait to see who Obama chooses.

"The one litmus test I have is, 'No ideologues,' " he said.