Davis will serve a 50-game ban for a second violation for a drug of abuse, the commissioner's office said Friday.
Morel was given a 25-game suspension for an unspecified violation. The only specified 25-game penalty in the minor league program is for unauthorized use or possession of a syringe.
Davis, 23, is on the roster of the Class A Augusta GreenJackets of the South Atlantic League but hasn't pitched this season because of a labrum injury.
Morel, 23, was 4-3 with a 4.33 ERA in 21 relief appearances this year for St. Lucie of the Florida State League and hasn't allowed a decision or given up a run during three games with the Class A Savannah Sand Gnats of the South Atlantic League.