The Teessiders are believed to be on the hunt for reinforcements in the middle of the park, having also been linked with Queens Park Rangers player Luke Freeman.
Saville has made 64 appearances for the Lions, scoring 10 goals and claiming three assists, since his move from Wolverhampton Wanderers last summer.
Wallace, meanwhile, has had two loan spells at the New Den from Molineux before signing permanently a year ago, and has scored 11 goals and registered 13 assists in 83 appearances.
According to Sky Sports News, the first initial bid for the midfield pair was £6m, but neither player has been considered for sale by the East London club.
Both Saville and Wallace featured in Millwall's 2-2 draw at home to Boro on the opening day of the Championship season.