Engel batted .224 with two home runs and 17 RBIs last season, his sixth with the White Sox. Mendick hit .289 with three homers and 15 RBIs in 31 games before tearing the ACL in his his right knee against Toronto on June 22. Payton was 3 for 21 over two stints with the White Sox last September.
All remaining unsigned players on the 40-man roster have been tendered contracts for next season.
Chicago missed the playoffs at 81-81 coming off back-to-back postseason appearances. The White Sox hired Kansas City Royals bench coach Pedro Grifol to replace Hall of Famer Tony La Russa as their manager nearly three weeks ago.