Monroe joined the Red Raiders about a week after running back Chadarius Townsend came over from Alabama.
Monroe played in LSU's 42-25 victory over Clemson for the national championship last season and recorded a tackle in the 63-28 win over Oklahoma in the semifinals.
A highly rated high school recruit out of the Houston area, Monroe had 21 tackles over three seasons for the Tigers, most of them as a redshirt freshman in 2017. His sophomore season was derailed by injuries.
Monroe has one year of eligibility remaining, while Townsend has two. They are set to join three other grad transfers in Michigan State linebacker Brandon Bouyer-Randle, offensive lineman Josh Burger of Wofford and Duke linebacker Jacob Morgenstern.