The deal is pending a physical.
Thornton, 37, figures to replace Logan as the Yankees' main lefty in the bullpen.
The Yes Network first reported the agreement.
Thornton was limited by injury, including an oblique issue, with the Boston Red Sox in 2013. He appeared in 20 games for Boston, throwing 15 1/3 innings.
He was traded to the Red Sox from the Chicago White Sox last season but was left off Boston's playoff roster. He finished the season with a 3.74 ERA in 60 appearances spanning 43 1/3 innings.
An All-Star in 2010, Thornton hasn't just been a pure lefty specialist during his career, as left-hand hitters have batted .233 against him, while righties hit .241.
The Yankees' plan is to have David Robertson slide into the closer role. They also have righties Shawn Kelley, Preston Claiborne and Adam Warren as holdovers from 2013. They could have a second lefty reliever in Cesar Cabral.
Hubble said that he and Thornton evaluated the market and felt the Yankees offered the best opportunity with a chance to win.