Drew, 30, missed the end of spring training and the start of the regular season after being hit in the head by a pitch and suffering a concussion. He played in four rehab games with the team's Double-A affiliate over the last few days.
Boston signed Drew to a one-year contract worth $9.5 million this offseason. He hit .223/.309/.348 in 79 games for the Diamondbacks and Athletics last year, and will give the righty-heavy Red Sox another left-handed bat.
Iglesias, 23, has gone 9 for 20 (.450) in the early going with five infield singles. His defense is his calling card, and he's expected to be the shortstop of the future at some point, likely beginning next year after Drew's contract expires.