Some Democrats expressed confidence that they could build public pressure on the Republicans to give Mr. Obama’s nominee a hearing.

“The idea of not even allowing a hearing strikes a chord that is pretty deep,” said Senator Chuck Schumer, Democrat of New York. “It will mount; it will get much stronger when the president has a nominee.”

But Republicans appear far more interested in mobilizing their base of conservative supporters than worrying about voters in the general election. Even if Republicans might appeal to moderate voters by giving Mr. Obama’s nominee a hearing, they reason, angering their own base would be more damaging.

Some allies of the president said they expected Republicans to hold firm, given the court’s crucial role and the intensity of feelings among conservatives. Mr. Axelrod predicted that Mr. McConnell and the other Senate Republicans would be “implacable” on the issue.

“I don’t think they are going to move,” he said. “Any Republican who breaks rank on this will face the full fury of the base.”

White House aides said the president would pick a nominee “in due time,” but they gave few hints about whom he was considering. Former top aides to Mr. Obama said they expected the White House to select someone who could overcome the Republican opposition.

“This is a lifetime appointment and an incredible part of his legacy,” said Stephanie Cutter, who helped guide the president’s previous nominees through the Senate confirmation process. Ms. Cutter said she expected Mr. Obama to avoid an obviously polarizing nomination in favor of a sitting judge who has had bipartisan support in the past.