Scotland international Gordon is available on a free transfer after leaving Sunderland in the summer.
Everton boss David Moyes admitted after Saturday's draw with Norwich that a lack of competition was harming Tim Howard's form at Goodison Park and he is now thought to be ready to bring in Gordon to add to his goalkeeping options.
Moyes is keen to bring in a new goalkeeper with Howard's deputy Jan Mucha out of contract at the end of the season and he has had several trialists in over the last few weeks to cast an eye over possible reinforcements.
Gordon joined Sunderland from Hearts in 2007 for £9million but his time at the Stadium of Light was blighted by a series of injuries.
A number of clubs have been linked with Gordon, but Everton are now ready to offer the keeper the chance to get his career back on track.