The Gateshead-born forward 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 Swansea impressed under Brendan Rodgers last season but he has seen his playing time restricted this term, with new manager Michael Laudrup favouring the impressive Spaniard Michu.
The 27-year-old has delivered when called upon, with four goals in his last five appearances reminding potential suitors of his quality, but he appears keen on a move away from the Liberty Stadium having fallen down the pecking order in attack.
It is understood Martin O’Neill has long regarded Graham as the preferred replacement for Louis Saha.
The Frenchman is believed to have flown already to South Africa for talks with an unknown club since his release by the Black Cats.
The former Manchester United forward failed to make an impact on Wearside since joining in the summer from Tottenham, and O’Neill has confirmed the 34-year-old’s time at the Stadium of Light has drawn to an end.
“We have told Louis he is free to speak to other clubs and to leave if he can fix himself up somewhere,” the Northern Irishman told the Shields Gazette.
“We have come to an agreement whereby we will cancel the remainder of his contract, shake hands and go our separate ways.”