The Toffees have been heavily linked with a move for the former Manchester United winger.
Everton's reported pursuit of Nani could be scuppered by the winger's current deal at Fenerbache, with Futbolarena claiming, via Teamtalk, that the Toffees have until just July 1 to secure a cut-price deal.
Nani has emerged as an alleged primary transfer target for the Merseyside club in recent weeks after impressing during his first season for the Turkish giants.
The Portuguese star netted 12 times in 46 matches in all competitions for his club, and has been impressive for his country so far at Euro 2016, scoring twice in four appearances to help them reach the quarter-final stage.
Even before his exploits in France the 29-year-old was thought to have caught the eye of Everton, and the Premier League club reportedly submitted a bid for the former Manchester United man earlier this month.
However, it appears as though they could be in a rush to get the deal completed, as Futbolarena report, via Teamtalk, that Nani has a release clause of £7 million in his contract which expires on 1 July.
That fee would sound a relatively reasonable price for a man who appears to still have what it takes to shine on a major stage, but if the report is true then once that deadline passes the Toffees could find themselves held to ransom.
With Nani impressing at Euro 2016, there are likely to be more interested parties surfacing, and if Everton want to complete a deal then they must act this week.
However, the first reports of their interest in the player emerged before Ronald Koeman was appointed as their new manager, and it is unknown whether the Dutch boss would be keen to persist with the pursuit or if he will switch his focus away from the Portuguese star.