The 27-year-old has struggled for game time with the Swans under new manager Michael Laudrup, leading to speculation that he may opt to leave the club during the January transfer window.
Sunderland manager Martin O’Neill, who is reportedly looking to bolster his attacking options, has outlined the former Watford man as one of his top targets and could press ahead with a move in the coming weeks.
Graham has recently returned to the South Wales club’s first-team, capping some impressive displays with a late equaliser against Arsenal in the FA Cup, forcing a replay.
The likes of Newcastle, Reading and Norwich have all been linked with a move for the front-man, but it is believed that the Black Cats are in pole position should he leave the Liberty Stadium.
According to reports they have already made an official enquiry, and are likely to follow this up with a concrete bid.
Sunderland have been far from prolific in front of goal this term netting on only 29 occasions in their 21 Premier League games, making a centre-forward a must this month.
O’Neill is keen to give Steven Fletcher some much needed support, and sees Graham as the perfect man to work alongside the Scot.