Rumbelow exited Saturday's game against the White Sox with a cut on his right thumb, Ryan Divish of The Seattle Times reports.
The 26-year-old reliever allowed three hits and an earned run in two-thirds of an inning. He is competing for a job in the Mariners bullpen after being dealt from the Yankees in November. The injury seems unlikely to have a major impact on his chances of earning a big-league job.