With Mr. Biden facing plunging poll numbers and turning 82 the month he’d be on the ballot, and Vice President Kamala Harris plagued by flagging poll numbers of her own, conversations about possible alternatives are beginning far earlier than is customary for a president still in the first year of his first term.

None of the prospects would dare openly indicate interest, for fear of offending both a president who, White House officials say, has made it clear to them that he plans to run for re-election and a history-making vice president who could be his heir apparent. No president since Lyndon Johnson in 1968 has opted not to run for re-election.

Still, a nexus of anxious currents in the Democratic Party has stoked speculation about a possible contested primary in two years. On top of concerns about Mr. Biden’s age and present unpopularity, there is an overarching fear among Democrats of the possibility of a Trump comeback — and a determination that the party must run a strong candidate to head it off.