Graham, who scored Swansea's equaliser in their FA Cup clash with Arsenal on Sunday, has been linked with a move away from the Liberty Stadium after finding his first-team chances limited this season.
The 27-year-old has started less than half of Swansea league games so far this term following the arrival of Michael Laudrup as manager.
The news has alerted a number of clubs with the likes of Newcastle, Reading and Norwich all linked with a move for Graham.
However, Sunderland have emerged as the front runners to land Graham during this month's transfer window.
Sunderland boss Martin O'Neill is keen to bolster his attacking options with his side struggling for goals this season and he is ready to make a move for Graham to help ease the burden on Steven Fletcher.
The Black Cats are thought to have made an enquiry to Swansea to ask about Graham's availability and they are ready to firm up their interest with a bid for the former Watford man.