WASHINGTON — Democratic leaders are convinced that Steve Bullock, the popular governor of Montana, would give the party its best shot at unseating Senator Steve Daines, a freshman Republican, when he stands for re-election next year in a state that President Trump carried by 20 points.

But Mr. Bullock has a bigger — if far less attainable — aspiration: running for president.

And so it has gone for Democratic leaders as they struggle to recruit a solid slate of candidates that they will need to net the three or four seats necessary to take control of the Senate next year. Four top-tier potential Democratic Senate candidates — John Hickenlooper in Colorado, Stacey Abrams in Georgia, Beto O’Rourke in Texas and Mr. Bullock — are seriously exploring presidential campaigns, forsaking statewide campaigns within their grasp, at least for now, for a national one that would be the longest of long shots.

A fifth potential recruit, Richard Ojeda, resigned his State Senate seat in West Virginia to run for president, only to withdraw last month — without committing to challenge Senator Shelley Moore Capito.

“This isn’t a bad map like 2018 because we have very few seats to defend,” said Guy Cecil, the former executive director of the Democratic Senatorial Campaign Committee. “But the opportunities to pick up seats are limited, and we need the best candidates possible to win the majority. It’s critical that some of the long shots in the presidential primary consider the Senate this year.”