The 28-year-old is to undergo a medical with the Lions today.
Camberley-born Saville - who has a year left on his deal on Teesside - moved to Boro almost three years ago in a £8m move in August, 2018.
Boro had initially tried to make it a double deal and also bring Jed Wallace to the club.
Former Chelsea academy player Saville played 118 matches for Boro and netted 12 goals, with getting him off the wage bill likely to provide Neil Warnock will more room for manoeuvre in the transfer market as he aims to bring in a number of new signings.
Boro have reportedly made a move to sign Wycombe striker Uche Ikpeazu, with the Teessiders set to pay around £1m for the 26-year-old, with the fee based on appearances and bonuses.
Talks have taken place over the past few days and Boro poised to make a breakthrough in their search for a new centre forward.
South American reports are also strongly linking Boro with Argentine midfielder Martin Payero, 22.