The Swans are keen to do a deal for the striker and have made an offer that is close to Stoke's valuation but Potters boss Pulis says any move will hinge on what the player wants.
Stoke are under no pressure to sell and despite reports to the contrary, Jones is still in the plans of Pulis and could stay at the Britannia Stadium, at least until the summer.
"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," Pulis said. "There is an offer there and it's around about what the club would expect, but I've said to the player if you don't want to leave mate, then you can stay, and he wants to stay.
"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. Because if he says no, there will be no problems with me."
A deal is by no means dead though and Swans boss Michael Laudrup will be confident of persuading him to jump ship should the two clubs agree a fee.