The Merseyside-based club have made significant additions to their squad in James Rodriguez, Abdoulaye Doucoure and Allan, but want to bring in a new centre-back before the transfer window closes.
Everton are considering a number of centre-back options after losing out on the signing of now Arsenal star Gabriel and talks to take Chelsea man Fikayo Tomori on loan stalled.
One of the players the Toffees are looking at is 22-year-old Godfrey and they have touched base with Norwich to enquire about a deal for the player.
However, Everton are only interested in a loan deal for the central defender, while Norwich want to sell him on a permanent basis; a stance which is proving to be a hurdle in the transfer.
Everton are reluctant to sign Godfrey on a permanent deal and Norwich value the defender at £25m.
With other options on their list, Everton are willing to end their interest in the player if Norwich do not agree to compromise deal.
A possible compromise deal could see the Toffees sign Godfrey on an initial deal followed by an option to buy in the future.