Top Senate Republicans vowed Wednesday to continue blocking President Obama’s nomination to the Supreme Court ahead of November’s presidential election, even if Obama chooses the Republican governor of Nevada to replace the late Justice Antonin Scalia.

Brian Sandoval, a centrist former federal judge who has served as governor since 2011, is among the Supreme Court candidates under White House consideration, according to two people familiar with the selection process. Some key Democrats view Sandoval as perhaps the only nominee President Obama could select who would be able to break a Republican blockade in the Senate.

But after The Washington Post published news of Sandoval’s consideration Wednesday, GOP leaders insisted that Obama nominating a Republican would make no difference.

Senate Majority Leader Mitch McConnell (R-Ky.), who pledged “no action” on any Supreme Court nomination before November’s election on Tuesday, said in a statement that the nominee “will be determined by whoever wins the presidency in the fall.”

The No. 2 Senate Republican leader, Majority Whip John Cornyn of Texas, said likewise: “This is not about the personality.”

White House press secretary Josh Earnest told reporters Wednesday he would not comment specifically on whether the administration was considering Sandoval because he did not want “to get into a rhythm of responding” to every report on a potential nominee. But he said that Obama was committed to finding “the best person to fill the vacancy at the Supreme Court,” regardless of party....