The affliction made things difficult with Downs' pitching motion, and officials put him on the DL backdated to Aug. 3, the day after he last pitched at Oakland.
Downs, signed as a free agent on July 6, has made 12 relief appearances for the Royals, posting a 3.48 ERA and receiving one loss. He began the season with the White Sox, pitching in 38 games.
Willingham was obtained on Monday from the Twins in exchange for Triple-A pitcher Jason Adam.
Dick Kaegel is a reporter for MLB.com. Jackson Alexander is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.