The Edmund clash will happen if former Wimbledon champion Murray comes through his opening test. Edmund, who has replaced Murray as British number one, receives a bye to round two.
Murray has been drawn to face American Mackenzie McDonald in the first round as he steps up his recovery from hip surgery.
It will be Murray’s first tournament since deciding not to play at Wimbledon.
He took part in the Queen’s Club and Eastbourne events earlier in the grass-court season, losing to Edmund at the latter, but a reunion is on the cards on the hard courts of the US capital providing Murray overcomes a player who reached the fourth round of Wimbledon.
That was 23-year-old McDonald’s best grand slam performance.
Murray has been warming up for his return in the company of former England football captain Rooney, who joined him on court for a lighthearted session.
Rooney is now Washington-based after leaving Everton to join DC United this summer.
He demonstrated his racket skills before joining in a game of foot tennis, with Jamie Murray also getting in on the action.
Germany’s Alexander Zverev is the top seed in Washington, with Wimbledon semi-finalist John Isner also taking part.
Andy Murray is targeting a return to grand slam action at next month’s US Open, having missed 11 months of action between last year’s Wimbledon and the recent grass-court campaign to get his hip in shape for competition.