Graham 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 led to a £3.5million move to Swansea following their promotion to the 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.
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 Sunderland manager Martin O'Neill has now tabled an offer.
Should the deal for go through, it is understood the Black Cats would be prepared to listen to offers for Campbell, who is thought to be attracting interest from Championship pacesetters Cardiff and Hull.
The bid for Graham comes as 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.
O'Neill told the Shields Gazette: "We have told Louis he is free to speak to other clubs and to leave if he can fix himself up somewhere.
"We have come to an agreement whereby we will cancel the remainder of his contract, shake hands and go our separate ways."
O'Neill added that a move to South Africa was a possibility for the former Fulham, Manchester United and Everton striker.
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