The Union will host the New York Red Bulls in a round of 16 U.S. Open Cup game on June 29 at Talen Energy Stadium. Also in the round of 16 draw that took place on Thursday, the New England Revolution will visit the NASL's New York Cosmos in the other East Region game.
If the Union wins, it would travel to the winner of the New England-Cosmos game on July 20.
MLS teams didn't begin competing until the fourth round, and the Union opened with Wednesday's 3-2 win over the visiting Harrisburg City Islanders.
Last season, the Union eliminated the Red Bulls in penalty kicks during an Open Cup quarterfinal.