Everton have turned down a request from Sunderland to take Phil Jagielka as part of a deal for Lamine Kone.
Club captain Jagielka turns 34 next week and came back from a hamstring injury to play 45 minutes against Espanyol at the weekend but is a doubt to start against Tottenham for their season opener.
Everton will complete the signing of Ashley Williams imminently, following the sale of John Stones, but Jagielka also has interest from other clubs and may weigh up his options before the transfer window closes.
Stones made the switch to the Etihad Stadium for a £47.