The Red Sox are thin at the position, since newly signed Mike Napoli (who primarily has been a catcher during his career and comes with hip issues) and Mauro Gomez (15 career major league games started at first) are the only first basemen on the roster.
The 36-year-old Overbay would fit well on the Red Sox bench, which is in need of another left-handed bat and a competent defensive backup at first base.
He has struggled offensively over the past two seasons after hitting 20 homers for the Blue Jays in 2010. Since then, he's bounced from the Jays to the Diamondbacks to the Braves, performing at a mediocre offensive level at each stop.
His best season came in 2006 with the Blue Jays, when he hit .312 with an .880 OPS and 22 homers. The Brewers reportedly also had been interested in Overbay.