The report claims that the Toffees new boss Ronald Koeman is keen on bringing in the former United winger.
Nani joined Fenerbahce from United on a three-year deal last season but is expected to leave this summer after falling out with head coach Vitor Pereira.
The Portuguese international has years of Premier League experience having won the league title four times and lifted the Champions League with the Red Devils.
United are also reported to be in the race to sign DHA with Turkish media outlet DHA claiming that personal terms have already been agreed.
Nani does have an €8 million release clause which is active only for Premier League, La Liga and Serie A sides.
Nani is currently in France on international duty with Portugal ahead of Euro 2016.