The deal is contingent on the injury-prone Drew passing a medical.
"I think Drew is a great fit to complement our existing personnel," Red Sox general manager Theo Epstein told the team's website.
"If you look, last year and the last couple of years -- especially last year -- a couple of our weaknesses were a number five hitter and outfield defense, generally, especially with a fly-ball staff, which we have.
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"Drew, if we end up signing him officially, addresses both of those areas of concern."
A combination of solid defence and versatile hitting, Drew batted .283 last season with the Los Angeles Dodgers, smacking 20 homers and driving in a career-high 100 runs.
In nine Major League seasons, Drew has also played for St Louis and Atlanta, posting a career .286 batting average with 162 homers and 509 RBI.
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