“When unions are strong, families are strong — and when families are strong, America is strong,” she often says.

But two major unions, the Service Employees International Union and the American Federation of State, County and Municipal Employees, have said they will hold off on an endorsement until they have completed the process of consulting with their members. Union leaders have said the decision has as much to do with Mr. Sanders as with Mr. Biden’s looming decision about whether to enter the race.

Against this backdrop, the possibility of losing formal support from the firefighters comes as a particularly unwelcome blow.

The Clinton campaign had been in touch with the union about the possibility of an endorsement during the first several months of 2015. But the talks became serious in June, when Mr. Schaitberger, with the assent of his board, told the campaign the union was strongly leaning toward endorsing Mrs. Clinton, pending the board’s formal vote.

The two sides later began working out details of the endorsement that Mr. Schaitberger and his board had viewed as nearly inevitable, including discussions about the timing and location of an announcement. The most recent plans would have had Mr. Schaitberger endorse Mrs. Clinton this month in Iowa.

In late July, senior union officials traveled to Mrs. Clinton’s campaign headquarters in Brooklyn to meet with several of her political and policy aides. The meetings took place under the assumption that the endorsement was on track, according to the union official. Their purpose was to educate the campaign about the policies that union members would want to see Mrs. Clinton discuss, and the rhetoric they would want her to use when discussing them.