According to Turkish news outlet Ajansspor, the three Premier League clubs, alongside Bundesliga side Werder Bremen and Spanish club Real Sociedad, have all submitted offers for the Turkish international.
However, Fenerbahce are keen to hold on to the 26-year-old until at least the end of UEFA Euro 2020 to allow his value to go up as much as possible.
Should Tufan perform well in the summer, his valuation could soar, with the report claiming that Fenerbahce are hoping to receive €20 million for the 58-cap Turkey international.
According to the report, Crystal Palace are currently leading the race to sign him, with the Eagles submitting an improved offer of €15 million from their earlier €13 million bid.
Both Crystal Palace and Fenerbahce are currently negotiating in Turkey, and they are hopeful of reaching a deal over the coming weeks.
However, Ajansspor are also reporting that negotiations with Everton and Leicester are very 'tight' as things stand.
The report adds that Crystal Palace have been in negotiations with Fenerbahce over Tufan for quite some time, and they have made 'extensive efforts' to try to sign him.
If his play style is anything like that of Toure, then he will go on to thrive in the Premier League, for whichever side he ends up going to.