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Hours earlier, Boston put center fielder Jackie Bradley Jr. on the 10-day disabled list because of a sprained left thumb.
Davis is expected to join the Red Sox in Cleveland on Thursday for a game between division leaders.
Red Sox president of baseball operations Dave Dombrowski said he began discussions with Oakland general manager Billy Beane last week about Davis.
He led the American League with 43 steals last year for Cleveland, and also hit a big home run in Game 7 of the World Series off Cubs closer Aroldis Chapman.
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