Kirkman, a left-hander, will undergo treatment for the recurrence of cutaneous lymphoma in the near future.
The lymphoma was discovered during a recent routine examination of Rangers players for signs of skin damage caused by exposure to the sun, The Dallas Morning News said.
Kirkman, who underwent treatment for the condition last year, is 0-2 with an 8.18 ERA in 25 relief appearances this season. He is expected to make a complete recovery, but the team said it was unclear when he might be back in action.
His spot in the bullpen was filled by right-hander Kyle McClellan, who compiled a 2.00 ERA in nine minor-league appearances.
The Rangers also place pitcher Alexi Ogando on the 15-day disabled list Sunday, due to right-shoulder inflammation. Ogando, 4-2 with a 2.93 ERA in 10 starts this season, just came back from a 10-day stint on the DL due to right biceps tendinitis.
Celebrity Families of 2014 [PHOTOS]