An official announcement of his signing could come in the next few days or early next week.
The two sides agreed to terms on a one-year, $750,000 contract last weekend, according to a source. Greene, 30, would become the club's utility infielder, replacing Omar Vizquel.
Greene played just 77 games with the St. Louis Cardinals last season because of two stays on the disabled list with social anxiety disorder. He hit .200 with six homers and 24 RBIs in 193 at-bats and lost his starting job.
The previous season, while with the San Diego Padres, he was limited by a broken left hand, suffered when he punched a storage chest near the dugout.
Before addressing his personal issues, Greene played well for the Padres at shortstop. He batted .254 with 27 homers and 97 RBIs in 2007, his best season in the majors.