The 27-year-old has started just 10 league games under Michael Laudrup this season - with star summer signing Michu taking centre stage.
Recent weeks have seen Graham getting a run of games, but it is understood that the striker would like to move back to his native North East.
The Gateshead-born front-man came through the youth ranks at Middlesbrough, but he failed to become a first-team regular at the Riverside.
Moves to Carlisle, Watford and then Swansea followed and it is now believed Graham would like a move closer to his roots.
Sky Sports understand Sunderland are very keen, whilst his boyhood club Newcastle have also been linked.
Source: Newcastle United Mad