At a fund-raising reception in New York City for Senate candidates last month, Mr. Schumer revealed that he was traveling to Montana and projected confidence that Mr. Bullock would agree to run, according to an attendee.

More quietly, a handful of union officials have reached out to Mr. Bullock to assure him he would have all the resources he needs in a state that still retains a strong labor tradition, according to one of the party officials, who was familiar with the conversations.

“I’m hopeful,” said Randi Weingarten, the president of the American Federation of Teachers, the powerful educators’ union, about the prospect of the governor entering the race.

What may have proved most decisive for Mr. Bullock, however, is that his family is supportive of his candidacy, according to the officials familiar with the discussions. He has long been reluctant to commute to Washington because of the toll he feared it would take on his family.

While his party has been avidly pursuing him, Mr. Bullock is not the only Democrat interested in the race. A handful of candidates are already running, including Mayor Wilmot Collins of Helena.

But after Mr. Bullock’s announcement on Monday, Cora Neumann, an economic development executive who had already raised over $650,000 for her bid, said she would withdraw from the race and threw her support behind the governor.

“I know that what matters most is that we come together in the effort to beat Senator Steve Daines,” she said in a statement, adding that Mr. Bullock “offers us the best chance to defeat Daines and flip the Senate.”