Talks started last season and United offered him a new deal in September, but that remains unsigned with negotiations on what will be the last big contract of the Belgium international's career at an impasse.
United are keen for the Belgium worldwide to stay but negotiations, which began last season, have dragged on after he rejected an offer in September.
United remain hopeful that they can still reach an agreement with him.
Foreign clubs can agree pre-contract terms with Fellaini from January, with Besiktas 4/7 favourites with Sky Bet to land the midfielder before the end of the January window.
Fellaini has emerged as a key part of Mourinho's team, after he endured a hard start when signing for David Moyes for £27.5m in the summer of 2013.
In total, 56 of Fellaini's 142 United appearances have come in the 15 months since Mourinho took charge and the manager has been a staunch supporter of the Belgian amid much conjecture from supporters and critics.
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