The news was first reported by NFL reporter Adam Caplan.
A versatile veteran, Klein started 44 games for the Saints over the past three seasons, including the playoffs. He first played as the Saints' every-down middle linebacker and signal caller in 2017. He played about 75% of their snaps on the strongside in 2018 and 2019.
His numbers have been remarkably consistent: 54 tackles and two sacks in 12 games in 2017; 70 tackles and two sacks in 2018; 69 tackles and 2.5 sacks in 2019. He had one interception in each of the past two seasons, including his first career touchdown last season.
Klein, who turns 29 on July 30, began his career as a fifth-round draft pick with the Panthers in 2013, where he started a total of 23 games in four years while backing up Luke Kuechly and Thomas Davis.
He has 339 tackles, 10.5 sacks and 6 forced fumbles in seven seasons.
ESPN's Marcel Louis-Jacques contributed to this report.