The Royals claimed Grossman off waivers from Texas on Saturday, Anne Rogers of MLB.com reports.
The Rangers placed Grossman on waivers Thursday, Kiley McDaniel of ESPN.com reports.
Grossman is out of the lineup for Monday's game against the Cardinals.
Grossman started at designated hitter and went 2-for-4 in Thursday's 2-1 win over the White Sox.
Grossman went 1-for-1 with a solo home run, a walk and an additional RBI in Sunday's 13-2 win over the Rays.
Grossman is out of the lineup for Wednesday's game against San Diego.
Grossman is out of the lineup for Tuesday's game against the Padres.
Grossman started in left field and went 1-for-2 with a walk, a stolen base and two runs scored in Wednesday's 5-3 win over the Mets.
Grossman started at designated hitter and went 0-for-3 in Monday's 2-1 loss to Detroit.
Grossman went 2-for-4 with a walk, a double, two RBI and a run scored in Saturday's 7-0 win over the Marlins.