The Gunners made the offer to take the 28-year-old to The Emirates which was rejected out of hand.
It is understood the ?10million offer was ?2million lower than the one made 12 months ago.
The Merseyside club view the bid as being well below the player's market value, particularly as the defender signed a new four-year contract in March.
Everton have yet to make any signings so far this summer and their tight finances means manager David Moyes is likely to have to sell players in order to be able to bring in new faces.
But Jagielka is not one of those on the list which the Toffees boss is looking to offload, although should a bid closer to the club's valuation of about ?20million come in they would have to seriously consider it.
Arsenal have also been linked with Bolton centre-back Gary Cahill and Blackburn's Chris Samba and Gunners boss Arsene Wenger could well explore those options.
Source: Team Talk
Source: Team Talk