For years, liberal Democrats have fought against proposals to cut Social Security benefits. Now, they’re pushing the party not just to defend benefits but to increase them, and that could present a problem for Hillary Clinton.

The call for higher benefits is a marked difference from recent years in which the White House and Republicans were negotiating deficit-cutting deals, leaving liberals to argue merely for staving off benefit cutbacks. Separately, many experts in both parties have long argued that extending the solvency...