As the nation’s senior Democrat, President Barack Obama is expected to work hard for his party’s congressional candidates. But in Florida’s unusual three-way Senate contest, the White House faces a complicated calculation.

White House officials have taken calls in recent days from at least three Democrats seeking to start a conversation about the candidacy of Florida Gov. Charlie Crist, the one-time Republican star who is now running for Senate as an independent. These Democrats are active in party politics but are nonetheless backing or leaning toward Mr. Crist in the Senate race.