Drew, who spent one season in Boston, turned down a $14.1-million offer from the Red Sox last year after helping the team capture their third title in 10 years.
The deal he accepted Tuesday, which is subject to his successfully passing a physical, has a $14-million salary. If he passes the physical and is added to the 40-man roster Wednesday, he would receive a prorated amount of $10,021,858.
The 31-year-old might require a minor league stint before joining the Red Sox. Once he is ready, Drew will take over at shortstop and rookie Xander Bogaerts will be shifted to third base.
Philadelphia Phillies ace Cliff Lee underwent an MRI exam on his sore throwing elbow, raising concerns the left-hander might become the latest major league pitcher to be sidelined by a serious injury.
Lee remained in Philadelphia to undergo the test after pitching 6 2-3 innings in a win over Cincinnati on Sunday. He has experienced mild elbow discomfort for the last three weeks, General Manager Ruben Amaro Jr. said.
The Phillies awaited test results as they began a series in Miami.
The Mariners put Hart on the 15-day disabled list before Tuesday night's game at Texas. They recalled infielder-outfielder Nick Franklin from triple-A Tacoma.
The 32-year-old Hart injured his leg swiping second during the fourth inning of the Mariners' 6-2 victory at Minnesota on Sunday. He has appeared in 37 games, hitting .209 with five home runs and 17 batted in.