The 27-year-old began his career with Boro when he joined their renowned youth academy in 2003.
Now whilst rivals Sunderland continue to talk terms with Swansea, Boro have come in and reached an agreement with the South Wales outfit.
He went on to play more than a dozen first-team games for Middlesbrough before moving on to Carlisle, Watford and then Swansea in 2011.
Graham has already told Swansea he would like to move away from the Liberty Stadium and a move back to the North East was always on the cards.
It is understood that a deal worth around £5million has been agreed between Boro and Swansea.