Senator Chris Murphy of Connecticut, who campaigned alongside Mrs. Clinton in his state, said she must focus on how best to connect to a broader electorate that is in what he called a “revolutionary mood,” alluding to how Mr. Sanders has energized supporters. “You are not going to win this general election by proposing incremental changes,” Mr. Murphy said, adding that he hoped Mrs. Clinton “doesn’t shy away from proposing some big ideas to try to reorder the country, to the benefit of those that are hurting.”

Mr. Sanders’s campaign vowed Tuesday to remain in the fight through the California primary on June 7, and perhaps until the party’s convention in July. As word of Mrs. Clinton’s victories rolled in on Tuesday night, he gave a defiant speech in West Virginia, predicting he would win the state when it votes May 10.

Mrs. Clinton, who doggedly remained in the 2008 primary contest against Barack Obama until June despite his insurmountable delegate lead and Democratic calls for her to bow out, has said she will not call on Mr. Sanders to withdraw, but she has bristled at the implication that he could overcome her big leads in both pledged delegates and the popular vote.

Mrs. Clinton’s aides and allies are pressuring Mr. Sanders to run on issues rather than continuing to attack Mrs. Clinton’s ties to Wall Street or her previous support for global trade deals — attacks that Republicans are likely to use in the fall.

“If he tones down the rhetoric and continues to fight, he’ll go out on a very high note with a lot of people, including me, thinking he did a great service to American democracy,” said Edward G. Rendell, a former Pennsylvania governor, who supports Mrs. Clinton. “But if he keeps it up, it could be brutal.”