Zimmer, the fifth overall selection in the 2012 MLB Draft, posted a 3-5 record with a 2.39 ERA combined between two levels last season. Almonte, 22, posted a 6-6 record and a 4.51 ERA split between two Minor League stops and also made nine relief appearances for the Royals last season.
Starling, the club's No. 7 prospect and the fifth overall selection in the 2011 MLB Draft, batted .254 with 10 home runs in 91 games for Northwest Arkansas last season.
The club also reassigned pitchers Christian Binford, Jonathan Dziedzic, Aroni Nina, Brooks Pounders and Sam Selman to Minor League Spring Training.