Remy has a couple of options in France but confirmed earlier this week that his preference was a move to the Premier League.
West Ham, Liverpool and Tottenham have all been tracking the 23-year-old but Stoke jumped to the front of the queue after their chairman Peter Coates and coach Mark O'Connor watched him in action at the weekend.
Tony Pulis flew to France on Wednesday to sell the club to Remy and it looked as though a quick deal could be on the cards when the France star travelled to England the following day to view Stoke's training facilities.
Remy eventually left without putting pen to paper, though, and speculation suggests he will return to England next week to talk to at least one more Premier League club.
However, any interest club will have to match the fee put up by Stoke, who are also tracking Jon Walters, Kevin-Prince Boateng and Kenwyne Jones.
Ipswich have rejected their latest bid for Walters but are resigned to the transfer going ahead, while a £4million offer for Boateng is thought to be on the table at Portsmouth.
Jones, meanwhile, will be allowed to leave Sunderland once they sign Manchester United's Danny Welbeck and Stoke are in the boxseat to capture the Trinidad and Tobago forward.