The mediation period between the pilots and United were scheduled to end by June 15, but a deal between the two sides was not reached before the deadline.
The NMB has overseen the negotiations involving the pilots and United, but the AFL-CIO called Monday for the panel to offer arbitration in the case.
"Our belief is that despite the intense and prolonged effort in negotiations, mediation has not succeeded, and we believe a release will be the only trigger to a successful completion of the negotiation process," Trumka wrote.
United has reached deals with other groups of employees, such as Continental flight attendants, but a deal with pilots has remained elusive.
-This post was corrected June 19 at 9:34 p.m. from an earlier version that confused the National Mediation Board with the National Labor Relations Board.