The Turkish sports newspaper reports that the Reds are keeping close tabs on the Portuguese international who has an €8.5 million release clause option.
Nani has been linked with a move away from Fenerbahce after falling out with head coach Vitor Pereira but has revealed that he will not make a decision on his future until after Euro 2016.
The Portuguese international has hit the ground running in scoring twice in the tournament making his joint top scorer for his team alongside Cristiano Ronaldo.
Nani has years of experience playing in England having won four Premier League titles and the Champions League with Manchester United.
The 29-year-old still has another two-years left on his contract at Fenerbahce but the club will not be able to reject offers that activate his release clause.
Nani joined Fenerbahce on a three-year deal from Manchester United last season but the Red Devils have also been closely linked with their former winger and are reported to have agreed personal terms with Nani according to Turkish media outlet DHA.
Inter, Valencia and Atletico Madrid are also reported to be interested in the Portuguese international winger.