Martin O’Neill had launched a bid to take the 27-year-old Swansea striker back to his native North East.
QPR have now joined Norwich, Reading and Aston Villa as the other challengers for his signature, but Sunderland still firmly believe they will win the race.
Stoke rejected a Swansea request to take Jones on loan with a view to a permanent deal. They are now waiting for Liberty Stadium bosses to come back with an offer that meets their valuation.
Once Swansea have secured a replacement for Graham – and they remain hopeful it will be Jones – the former Watford star will be allowed to leave for around £4million, with the Stadium of Light his likely destination.