The report states that Fenerbahce have been monitoring the Costa Rica international and that the Istanbul club’s sporting director Giuliano Terraneo may travel to London in January to wrap up the deal.
The Milliyet go onto state that Fenerbahce are ahead on the race to sign Campbell due to their stronger financial position.
Campbell will have six months remaining on his current deal in January meaning clubs can hold talks directly with the player.
The Lions were closely linked with Galatasaray over the summer but Arsenal boss Arsene Wenger insisted that the 23-year-old forward stayed.
The 23-year-old has played just 398 minutes of football this season over the span of seven appearances.
The Istanbul giants currently lack options going forward and are looking to boost their attacking options over the winter transfer window.