WASHINGTON—The Obama administration upped the ante Wednesday in what is likely to become a protracted and heated battle over how to fill an election-year Supreme Court vacancy created by Justice Antonin Scalia’s death.

President Barack Obama said he won’t back down from picking a nominee, a direct challenge to top Senate Republicans who the day before said they won’t hold hearings or meet with any nominee Mr. Obama puts forward. At the same time, people familiar with the matter said the administration is vetting Brian Sandoval, the Republican governor of Nevada, as a possible candidate.

Mr. Sandoval—who initially on paper would seem hard for many Republicans to refuse—is far from the nomination. But the emergence of his name as a possible contender opens a new window into the administration’s thinking. The White House has already indicated that the president intends to nominate someone who has past Republican support and lacks a significantly liberal background.

Mr. Obama on Wednesday said he intends to pressure Republicans to change their minds by offering a nominee they would have difficulty refusing.

"I recognize the politics are hard for them,” Mr. Obama said. “I think the situation may evolve over time,” Mr. Obama added, saying that “I don’t expect any member of the Republican caucus to stick their head out at the moment and say that, but let’s see how the public responds to the nominee we put forward.”