The Toffees were unable to sign Dynamo Kiev forward Andriy Yarmolenko and are lacking options upfront after selling Steven Naismith to Norwich.
Galatasaray held talks with Lokomotiv Moscow over the summer and are still in the race to sign the 25-year-old according to HaberTurk.
Former Galatasaray manager Hamza Hamzaoglu is credited with discovering Niasse and putting him on the map during his time as Akhisar manager.
Hamzaoglu attempted to sign Niasse during his time in charge of Galatasaray and he is reported to have remained on the Lions shortlist even after his departure.
Niasse first made a name for himself in Turkey establishing himself as one of the most sought after strikers in the Super League before moving to Russia for €5 million.
The Lokomotiv striker has 12 goals and 10 assists in 21 games for the Russian giants this season.