The deal is for one year, league sources told ESPN.
Mack, who played for the Orlando Magic last season, will be starting his eighth NBA year.
The Grizzlies are working to strengthen the point guard position behind franchise star Mike Conley.
Mack's best pro season was in 2015-16 for the Utah Jazz, when he averaged 12.7 points and 5.3 assists. He was a second-round pick out of Butler in 2011.