Green-Ellis likely will replace Cedric Benson, who is a free agent, as the Bengals' featured running back.
The 26-year-old Green-Ellis has not turned the ball over in a span of four NFL seasons, which covers 53 games played and 510 rushing attempts.
Green-Ellis rushed for 667 yards and 11 touchdowns last season, sharing time in the Patriots' backfield rotation.
His best season came in 2010 when he rushed for 1,008 yards and 13 touchdowns.
He originally signed with the Patriots as an undrafted free agent out of Mississippi.
Information from ESPN AFC North blogger Jamison Hensley was used in this report.