It has been reported that the Toffees had agreed a £25million deal to sign the Turkey international to a four-and-a-half-year contract.
Tosun got on the scoresheet to help the Besiktas advance to the Turkish Cup quarter-finals with a comfortable 4-1 victory against Osmanlispor on Thursday.
After the match, claims the 26-year-old was heading for Merseyside were declared false.
"There is no offer for Cenk Tosun," said Metin Albayrak, spokesman for the Besiktas board of directors.
"We are only learning this from the press, and if an offer comes to us, we will decide."
Having only just signed Domagoj Vida from Dynamo Kiev and rumours of a move for Arda Turan persisting, Besiktas are looking to build on an impressive first half of the season, in which they progressed to the knockout stages of the Champions League.
In the Super Lig, they remain in touching distance of leaders Istanbul Basaksehir and Tosun has been an important contributor, leading the team with eight goals.
Besiktas coach Senol Gunes affirmed no deal is done regarding his side's star striker, who he hopes to retain.
"There is nothing official about transferring Cenk Tosun," said Gunes. "I am happy with my team and I do not want my players to go."