Sunderland’s chances of bringing Gateshead-born Graham back to the North East revolve around Swansea replacing the ex-Middlesbrough forward with Jones before the end of the transfer window.
For the domino-effect of deals to take place, Swansea must up their offer for Jones to around £6m after seeing a bid of £3m rejected for the Trinidad international last week.
Pulis: “Kenwyne has never come into my office and said he wants to leave the football club.
“It’s great speculation. If the two clubs are talking then they have to talk and speak to me and then I’ll speak to Kenwyne.
“If an offer comes in for any player in the Premier League and it’s the right offer then you look at it and then speak to the player.
“If it comes to a point where we accept an offer from Swansea then it will be down to Kenwyne whether he wants to go on that.”
Sunderland made a formal bid for Graham last week and Martin O’Neill will hope to make further progress before Swansea visit the Stadium of Light in nine days.
But Sunderland face competition for Graham’s signature, most notably from Norwich City, who have submitted bids both for the Swansea striker and Celtic frontman Gary Hooper.