Given Mrs. Clinton’s lead, several of her allies contended that he should shift his focus from criticizing Mrs. Clinton to championing Democratic policies that might help the party beat Donald J. Trump, Senator Ted Cruz of Texas or another Republican nominee, and also win back control of the House and the Senate.

“There does come a point when Democrats, and those who believe we can’t afford a Trump or Cruz presidency, should recognize that we need to start thinking about what’s best for the presumptive nominee going forward,” said Gov. Jack Markell of Delaware, a Clinton supporter whose state holds its primary on Tuesday. “The good news is that Bernie Sanders will remain in the Senate, where he can continue to speak out and continue to inspire new people into the process.”