The 24-year-old custodian, a £6 million signing from Ipswich Town 12 months ago, has failed to oust David Seaman between the Highbury sticks and has chosen to kick-start his career with a move to Goodison Park.
"For him it was not about money or anything. It was about playing - and hopefully he will go on to become the England keeper," said the player's agent Jonathan Barnett.
Toffees coach David Moyes turned to Wright after failing in a bid for Derby County's Mart Poom, but Everton chief executive Michael Dunford was delighted with the deal.
"We're delighted that Richard has agreed to join. And when everything is complete, he will travel to Scotland to join up with the first team squad," Dunford said.
Wright will cost an intital £3.5 million rising to £4.5 million depending on appearances, and is expected to pen a five-year deal on Thursday.
Meanwhile, Brazilian Juliano Rodrigo has passed his medical and will complete the formalities of a transfer to The Toffees.
"The player has now passed his medical and we expect to complete the transfer on Wednesday," said Dunford.
