Manager Arsene Wenger has been impressed by the England international who came back strongly last season from a cruciate ligament injury but Everton do not want to sell.
He joined from Sheffield United for £4m in 2007 and has been a pivotal figure in Everton's improvement since.
Driving force: Arsene Wenger wants Everton's Phil Jagielka to steer his Arsenal defence next season
Wenger, who is also keen on Everton's Steven Pienaar, needs two defenders. He will lose William Gallas and sees Sol Campbell as a squad player.
The Frenchman made an improved offer for Lorient's Laurent Koscielny last Sunday but is still waiting to hear if the £8.2m bid was successful.
Wenger is also interested in Fenerbahce's Andre Santos and Pape Diakhate of Dynamo Moscow.
In demand: Aston Villa and Spanish outfit Valencia are both keen on Arsenal striker Eduardo
Meanwhile, Aston Villa and Valencia are interested in Arsenal striker Eduardo.
The Croatia international, 27, is available at the right price with Arsenal keen to offload before bringing players, such as Joe Cole, in.
Arsenal boss Wenger plots swoop for Everton's Pienaar after the World CupCole's arrival at Arsenal could spell Eduardo exit as Villa eye Croatia ace West Ham told to improve £7m offer to secure deal for Everton striker YakubuARSENAL FC