The deal calls for a $9.5 million salary and $500,000 in performance bonuses, the sources said.
Drew, 29, is the younger brother of former Red Sox outfielder J.D. Drew.
Drew became a free agent when the Oakland Athletics declined their half of a $10 million mutual option for Drew in October. Drew received a $1.35 million buyout.
Drew, acquired by Oakland in an August trade with Arizona, batted .250 with five homers and 16 RBIs in 39 regular-season games for the A's.
Drew returned last season after missing 137 games dating to July 20, 2011, when he fractured his right ankle sliding into home against Milwaukee. He played in 40 games with the Diamondbacks last season and was hitting just .193 before the trade.
Like his brother J.D., Drew was a No. 1 draft pick out of Florida State, chosen 15th overall by the Diamondbacks in 2004. J.D. was in the big leagues two months after he signed; Stephen spent just one season in the minors before he made it, and by age 24 had become the team's everyday shortstop, playing an average of 147 games a season from 2007-2010. In the last three years of that span, his .800 OPS was exceeded by only two shortstops, Hanley Ramirez (.917) and Troy Tulowitzki (.883), and while he was considered average defensively, he made just 10 errors in 2010.
Red Sox general manager Ben Cherington has maintained throughout the offseason that shortstop isn't at the top of his to-do list, but also said the job will not just be handed to all-glove, no-bat prospect Jose Iglesias.
The 22-year-old Iglesias hit just .118 in 68 at-bats for the Red Sox in 2012. He was expected to be given an opportunity to earn the starting shortstop job in spring training. It is unclear if that's still the case, however, with the addition of Drew.
"We're just trying to be opportunistic on that," Cherington said at the winter meetings. "We feel Jose is ready to compete for the job. We're not ready to give it to him."
Recently, Iglesias was in Arizona training with Red Sox second baseman Dustin Pedroia.
Information from ESPNBoston.com's Gordon Edes and The Associated Press was used in this report.