A split is emerging inside the Bernie Sanders campaign over whether the senator should stand down after Tuesday’s election contests and unite behind Democratic front-runner Hillary Clinton, or take the fight all the way to the July party convention and try to pry the nomination from her.

One camp might be dubbed the Sandersistas, the loyalists who helped guide Mr. Sanders’s political ascent in Vermont and the U.S. Congress and are loath to give up a fight that has far surpassed expectations. Another has ties not only to Mr. Sanders but to the broader interests of a Democratic Party pining to beat back the challenge from Republican Donald Trump and make gains in congressional elections.

Mr. Sanders in recent weeks has made clear he aims to take his candidacy past the elections on Tuesday, when California, New Jersey and four other states vote. But the debate within the campaign indicates that Mr. Sanders’s next move isn’t settled.

For now, Democratic officials, fund-raisers and operatives are getting impatient, calling on Mr. Sanders to quit the race and begin the work of unifying the party for the showdown with the Republican presumptive nominee.

Orin Kramer, a New York hedge-fund manager who has raised campaign funds for both President Barack Obama and Mrs. Clinton, said with respect to Mr. Sanders’s future plans: “I would hope people would understand what a Trump presidency would mean and act accordingly—and ‘accordingly’ means quickly.”