The former Real Madrid midfielder moved to the Turk Telekom Arena on Friday, despite being linked with the Sukru Saracoglu club
The Turkey international was closely linked with both Istanbul giants after an unsuccessful season at the Bernabeu, with the 29-year-old opting instead to join the Lions for €3.5 million this week.
"We made a proposition to Hamit, but the transfer did not materialise. [Club executive] Ali Yildirim's statement explains the whole process," he stated a press conference.
"This is water under the bridge now. We wish Hamit all the best at Galatasaray."
Aykut went on to deny that he had made any attempt to sign Besiktas duo Manuel Fernandes and Mustafa Pektemek, and that any future arrivals depended on players leaving the club first.
"I have made no requests for Fernandes or Mustafa. If we have a player asking to leave, then I might consider transferring new players.
"In relation to players who will leave, we have made no decision concerning [Henri] Bienvenu."