The English forward is expected to leave Manchester United in the January transfer window, with Barcelona as the club he desires the most.
Despite being offered the forward, Fenerbahce have no plans to pursue a deal
"Actually, yesterday it was in the news that we are interested in him," Ilicali told Sky Sports.
"I think Rashford is a fantastic player, I really like him. But at the end of the day, when you have some players in your team (who play in the same role) it is not logical to get two players in the same position which are much better than other players.
"What I will say is that these types of players need to play, play, play. If he comes, then one of our players wouldn’t be able to play. When it came in front of us, unfortunately we said no for this transfer.
"There is nothing that we are interested in him. Of course, there are some rumours that spread. I can say very comfortably that we are not interested, but he is a good player."