Zimmer was selected by Kansas City in the first round, fifth overall, of the 2012 draft. The 27-year-old has dealt with injuries throughout his career, going 12-19 with a 3.54 ERA in 85 appearances, including 49 starts. He has totaled 310 strikeouts and 90 walks in 259 innings in the minors.
Under the deal announced Friday, Zimmer will earn $555,000 if he is in the majors next season, and $124,000 in the minors.
The 26-year-old Cuthbert hit .194 with three homers and seven RBI in 30 games with the Royals last season.