The Angels also recalled outfielder Jordyn Adams and infielder Kyren Paris on Friday ahead of their series opener against Detroit.
Moniak and Moustakas have a chance to return for the final week of the regular season before the moves are retroactive to Thursday.
Moniak has back tightness that has kept him out of the Angels' lineup since Sept. 5. The former No. 1 overall pick's production has slowed in recent weeks after his breakout summer, but Moniak is still batting .280 with 13 homers, 43 RBIs and a .797 OPS.
Moustakas has a left forearm strain. The 13-year major league veteran is batting .243 with eight homers and 30 RBIs in 61 games while becoming a fan favorite in his first chance to play in his native Los Angeles area.
Adams and Paris were both assigned to Triple-A Salt Lake on Thursday when the Angels recalled veterans Jared Walsh and David Fletcher. Both rookies have struggled at the major league level this season.
AP MLB: https://apnews.com/hub/MLB