KEENE, N.H.—Searching for an alternative to Hillary Clinton for 2016, some Democratic donors are meeting with potential challengers. Liberal activists are trying to coax Sen. Elizabeth Warren into running. Politicians not named Hillary Clinton are testing their appeal in New Hampshire and Iowa.

As formidable as Mrs. Clinton looks even before declaring herself a candidate, liberals are casting about for a committed populist to run against her in 2016. They see the former secretary of state and senator as too closely aligned with large corporations and question whether she can be counted on to narrow the income gap in America.

They hope to either recruit a candidate able to capture the nomination outright or at least give Mrs. Clinton enough of a scare that she embraces progressive policy goals. Their aim is to make the primary process a debate over the Democratic Party’s direction, rather than an uncontested march by Mrs. Clinton to the nomination.

Guy Saperstein, a Democratic donor and part-owner of the Oakland A’s baseball team, met privately at his home near San Francisco last week with Sen. Bernie Sanders, an independent from Vermont who has long championed liberal causes. Mr. Sanders says he is considering a presidential bid and wants to gauge whether he can raise enough money.

In their conversation, Mr. Saperstein said, he told Mr. Sanders that he couldn’t support him until he is assured Ms. Warren, of Massachusetts, won’t run. But he said he isn’t inclined to give money to Mrs. Clinton in any scenario, saying he is “extremely concerned” about what he called her “closeness to Wall Street.”