The striker, who scored 14 goals in the Championship last season despite only starting 20 times, has become a key target for Boro manager Garry Monk.
It now looks likely that the he will be on his way out of the City Ground this summer with Boro, Burnley and Hudderfield linked with to the player.
It has been reported that a £15 million offer has been lodged by Boro and are increasingly edging closer to finalise the move.
The 24-year-old began his career with Watford in 2011 before leaving for Peterborough United, where he scored 23 goals in 43 appearances.
Assombalonga has scored 30 goals in 47 Championship starts since a £5.5 million move to Forest in August 2014.
Speaking in June, Nottingham Forest boss Mark Warburton said “I don’t want Britt to go anywhere. He is a goalscorer.”
“He showed his value at the end of last season and we know what he can do.”
“Good players are sought after by other clubs, it’s as simple as that,” Warburton added.
"We are really clear that we don't want Britt to go anywhere, we are delighted with the work he is doing. But every player in the world market has a price on their head."