Senator Bernie Sanders of Vermont and former Vice President Joseph R. Biden Jr., two front-runners in the Democratic presidential race, are mounting a late push to gain a decisive advantage in the early primary and caucus states, aiming to avert a monthslong delegate battle against two insurgent rivals — and each other.

Mr. Sanders and Mr. Biden appear nearly deadlocked in the early states with Senator Elizabeth Warren of Massachusetts and former Mayor Pete Buttigieg of South Bend, Ind. Mr. Sanders held a slim lead in Iowa in a poll published Friday by The Des Moines Register and CNN, while recent polls in New Hampshire have found the leading candidates closely bunched. Mr. Biden has held a more consistent advantage in Nevada and South Carolina, the other states to vote in February.

In a sign of how extraordinarily fluid the race remains less than a month before voting, all four of the top candidates are in a position to win both Iowa and New Hampshire, an outcome that could transform the race for any of them.

Mr. Biden, the leading candidate in national polls, is especially determined to deny his challengers a chance to seize sudden momentum in the first few primary and caucus states. Hopeful of an early victory, Mr. Biden quietly dispatched his deputy campaign manager, Pete Kavanaugh, to move to Des Moines late last year to oversee his Iowa campaign. And in recent weeks Mr. Biden has made Iowa the most urgent priority on his travel schedule.