An MRI on Strasburg's shoulder revealed no damage, and the star pitcher will visit a neurologist in Los Angeles to evaluate a potential nerve issue, per Dan Kolko of MASN.
Washington's decision comes on the heels of a disappointing outing for Strasburg in his most recent start.
After allowing six earned runs on eight hits and two walks in 4.2 innings against the Atlanta Braves last week, Strasburg got into an argument with fellow starter Max Scherzer in the dugout.
The Nationals placed Strasburg on the DL in June with right shoulder inflammation. Injuries have been a recurring problem for the three-time All-Star, with ESPN.com noting he made six trips to the DL over the past four years prior to his latest stint.
Injury issues aside, Strasburg has been fairly solid in 2018. While not pitching at the standards he set over the last few seasons, he's 6-7 with a 3.90 ERA and a 3.50 FIP in 14 starts.
Strasburg finished with a career-best 2.52 ERA (minimum six starts) last season.
While Milone is expected to replace Strasburg for the time being, there will be additional pressure on Scherzer, Gio Gonzalez, Tanner Roark and Jeremy Hellickson to step up while Strasburg is out as well.
The Nats are seven games behind the Philadelphia Phillies for the NL East lead, and they trail the Atlanta Braves by 5.5 games for the final wild-card spot in the NL.
If Strasburg's DL stay is an extended one, an already difficult trek to the postseason will become even tougher for Washington.