After months of insisting he would not challenge Senator Steve Daines, Mr. Bullock, who ran for president last year, has told Democrats in the last week he is now inclined to run in what would immediately become one of the marquee Senate races of 2020. Mr. Bullock has only a few days to finalize his decision: the filing deadline to run in Montana is Monday.

In an email on Tuesday, Mr. Bullock, 53, declined to say whether he would run. The Democratic officials spoke on the condition of anonymity to discuss internal deliberations.

Senate Democrats are chiefly on the offensive this year, and have high hopes to defeat incumbent Republicans in Arizona, Colorado, Maine and North Carolina, where on Tuesday their preferred candidate captured the nomination. But with Republicans holding a 53-seat majority, and favored to defeat Senator Doug Jones of Alabama, it has not been clear which state could offer Democrats the fifth seat they would need to win a clear majority.