The Royals had interest in retaining Madson, who signed a Minor League deal with the team last spring and then made the club out of Spring Training. He appeared in 68 games for the Royals after missing the 2012-14 seasons while rehabbing from Tommy John surgery.
Madson had a 2.13 ERA with three saves during the regular season. He also appeared in nine postseason games and had a 5.40 ERA.
Madson was a pivotal part of the Royals' bullpen, serving as the primary setup man down the stretch to closer Wade Davis.
Kansas City still has right-handers Luke Hochevar and Kelvin Herrera to set up Davis, but Madson's departure will weaken its depth. Left-hander Tim Collins was signed to a 2016 deal recently, but he is coming off Tommy John surgery and likely won't pitch with the big league club until May.
The Royals also will have to fill the void left by Greg Holland, who had Tommy John surgery in October and was non-tendered. The Royals are open to bringing him back, however, to rehab with the team in 2016 and then pitch in '17.
The Royals have had encouraging dialogue with right-hander Chris Young about a possible return. General manager Dayton Moore described Young as possibly "the MVP of the staff last season."
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.