The Welshman is set to leave the Emirates for free in the summer.
The Gunners had offered the Welshman a new deal, but recently backpedal- and Toffees boss Marco Silva is now looking to snap him up.
Everton director of football Marcel Brands is putting a package together to tempt the 27-year-old to Merseyside.
The Wales midfielder will be out of contract and can command a stunning £15million signing-on fee.
If he agrees to join Everton, he could get a five-year deal worth £150,000 a week and become the main man in Goodison Park.
Ramsey has attracted interest from Liverpool as a long-term successor to James Milner.
Chelsea and Manchester United are also monitoring his situation