Police said Britton, 23, was arrested Saturday on multiple charges, including suspicion of drunk driving, after his truck left the road while he attempted to elude officers pursuing him.
The team released a statement saying, "This is a matter the Red Sox take very seriously, and is being addressed."
Britton was optioned to Double A Portland on Monday, and WFTX-TV, Fort Myers, reported his arrest that night.
Britton was charged with DUI, DUI property damage for striking a barbed-wire fence with his pickup truck and reckless driving. Police noted his slurred speech, bloodshot eyes and a strong odor of alcohol at his arrest, The Boston Globe reported.