The Express report that Everton are willing to activate his release clause while Manchester United who are also in the race want to negotiate a better deal – considering they sold him to Fenerbahce for less last summer.
Recently appointed Toffees boss Ronald Koeman wants to bring in the 29-year-old winger over the summer and Fenerbahce appear keen to part ways after Nani fell out with head coach Vitor Pereira.
United are reported to have already agreed personal terms with their former player according to Turkish media outlet DHA.
Italian giants Inter Milan and La Liga outfit Atletico Madrid are also in the race for the Portuguese international winger.
Nani has years of Premier League experience having won the domestic title four times and the Champions League during his time at United.
The Portuguese intentional scored scored 13 goals and provided 12 assists in 46 appearances for Fenerbahce last season.