WASHINGTON—The White House gave its clearest indication yet Thursday that President Barack Obama is looking to name a Supreme Court nominee who has past Republican support and lacks a significantly liberal background, as he approaches a major battle with Capitol Hill over filling the vacancy created by Justice Antonin Scalia’s death.

“The Senate gets to have a say, and so in order to get this done, the president is not going to be able to go out—nor would it be his instinct anyway—to pick the most liberal jurist in the nation...