Tom Henderson, Democratic chairman in Polk County, Iowa, said Mr. Biden is in danger of missing his moment and can’t afford to deliberate much longer.

“There are a number of people here in Iowa that are still waiting to find out if he’s going to run, who will commit to him if he runs,” Mr. Henderson said. “On the other hand, I don’t understand why there’s a continued delay in the decision. When you’re setting up a statewide organization, it’s really important to get that started.”

Eight days after the Iowa caucuses, New Hampshire holds the nation’s first primary. The Clinton family has a long history with the state’s voters. Her husband memorably dubbed himself the “Comeback Kid” after a strong second-place finish there in 1992. Mrs. Clinton won New Hampshire outright in 2008.

Mr. Sanders, who represents neighboring Vermont, presents yet another challenge as he has been drawing huge crowds. Mr. Sanders has 23 paid field staff working in New Hampshire. Recent polls show him beating Mrs. Clinton in the state, gaining traction by dint of a populist message that couples red-meat attacks on the “billionaire class” with calls for higher taxes on wealthy Americans.