Graham is reportedly unsettled at the Welsh club and is believed to be keen on a move back to the north-east.
The Gateshead-born 27-year-old began his career at Middlesbrough, before productive spells at Carlisle and Watford gave way to a £3.5million move to Swansea following their promotion to the Barclays Premier League in 2011.
Graham struck 12 goals in 36 league appearances as the Swans impressed under Brendan Rodgers last season, but he has seen his playing time restricted this term, with new manager Michael Laudrup favouring impressive Spaniard Michu.
Nevertheless, Graham has delivered when called on, with four goals in his last five appearances reminding potential suitors of his quality.
Reading Norwich and QPR have all been linked with the player in recent weeks, and amid the high levels of interest Sunderland manager Martin O'Neill has now tabled an offer.
Should the deal for go through, Press Association Sport understands the Black Cats would be prepared to listen to offers for Fraizer Campbell, who is thought to be attracting interest from Championship pacesetters Cardiff and Hull.
The bid for Graham comes as striker Louis Saha has been told he is free to leave the Stadium of Light.
The 34-year-old Frenchman arrived on a one-year deal in the summer after being released by Tottenham but has failed to make an impact.
- Sports & Recreation
- Barclays Premier League
- Brendan Rodgers