Chester-le-Street-born Graham had been expected to join Sunderland, who had bid around £5million for the 27-year-old.
Graham remains Sunderland’s No1 target after the departures of Fraizer Campbell and Louis Saha.
But Graham's potential move to the Stadium of Light depends upon Swansea bringing in another striker to take his place.
According to the Sunderland Echo, a Turkish club have entered the race for Jones, although the 28-year-old would rather stay in the Premier League.
The bookies have slashed the odds from 16/1 to 10/1 on signing Graham signing for Norwich City before the window slams shut.
But reports also are circulating, that Stoke are interested in the player to.
Stoke City are set to shake up the chase to sign the striker by offering an exchange deal with Kenwyne Jones.
With Swansea’s £3million bid for Jones already having been turned down by Stoke, the Britannia Stadium club are to propose a straight swap deal for the two strikers.