And they hope to sign the Hungarian international next week if a price can be agreed in the wake of Liverpool keeper Scott Carson joining West Brom instead of Stoke.
The Potters failed with a bid for 25-year-old Fulop last month – City believed to have been ready to go as high as £2million, but Sunderland wanting £2.5million.
But the Potters have come back in for a player who impressed on loan at Leicester City last season and who Roy Keane says he rates so highly, that he is no rush to let him leave.
City boss Tony Pulis is a big admirer of Fulop and agreed a loan deal for the keeper in January.
That move was cut short inside a week after Sunderland keeper Darren Ward picked up a back injury and Fulop had to be instantly recalled.
Having won an unexpected promotion, Stoke see Fulop as an ideal choice of keeper to strengthen them in the top flight.
Keane would love to keep Fulop at Sunderland but respects the fact that the keeper must be playing regular first-team football if he is to continue as his country's number one.
And Fulop's international career is of massive importance to him.
After deciding Fulop was too pricey, Stoke turned their attention to England keeper Carson, who was free to leave Anfield and Liverpool accepted a £3.5million bid for the player.
But that potential move has now collapsed and having been prepared to go higher for Carson, the idea of getting Fulop for a more modest fee has its appeal.