Barkley was a target for Tottenham, Chelsea and Arsenal during the summer window and turned down a deadline day move to Stamford Bridge due to his ongoing fitness concerns.
Spurs and the Premier League champions are understood to retain an interest, with Barkley in the final year of his contract at Goodison Park after rejecting an extension at the end of last season.
Barkley is expected to reassess his options in January, once fully fit again, and is believed to prefer a move to Spurs to work under Mauricio Pochettino.
Should the 23-year-old play for Everton then move to Champion League contenders Spurs or Chelsea in January, the midfielder will be cup-tied in European competition if either club drop into the Europa League after the group stage. Barkley cannot play for two teams in the same European competition.
Uefa rules state: “One player from the above quota of three [of new additions before knockout rounds] who has been fielded in a UEFA club competition group stage match for another club in the current season may exceptionally be registered, provided that the player has not been fielded: a. in the same competition for another club; or b. for another club that is currently in the same competition.”
Tottenham have a daunting Champions League group of Real Madrid, Borussia Dortmund and APOEL. Chelsea also face a tough task with Atletico Madrid, Roma and Qarabag.
Everton have been drawn with Lyon, Atalanta and Apollon Limassol in Group E of the Europa League.