[2] Klein was selected by the Kansas City Royals in the fifth round with the 135th overall pick of the 2020 Major League Baseball draft.

[7] MLB Pipeline named Klein the Royals' Pitching Prospect of the Year following the season's end.

[9] He returned to play in mid-May with the Northwest Arkansas Naturals of the Double–A Texas League as a member of their starting rotation, but was later moved to the bullpen.

[19] On July 30, 2024, the Royals traded Klein, Mason Barnett, and Jared Dickey to the Oakland Athletics in exchange for Lucas Erceg.

[21] On January 21, 2025, Klein was traded to the Seattle Mariners in exchange for international bonus pool space.