CHARLESTON, W.Va. — Senator Joe Manchin III is just doing his homework.

That’s how Mr. Manchin, Democrat of West Virginia, described a round-table discussion about President Trump’s Supreme Court nominee, Judge Brett M. Kavanaugh, with about two dozen local and state leaders on Friday.

The round table, and Mr. Manchin’s careful contemplation of his decision, underscore the tightrope being walked by Democrats running for re-election in states Mr. Trump carried in 2016.

If they vote to confirm Judge Kavanaugh, senators like Mr. Manchin, Joe Donnelly of Indiana and Heidi Heitkamp of North Dakota risk fracturing the party. But if they vote against him, they risk jeopardizing their seats in November’s midterm elections — and a shot at a Democratic majority.

Mr. Manchin, who is a pivotal piece of the Democratic leadership’s uphill bid to block the nomination, gave few indications during the meeting of how he would ultimately vote.