O'Kane, who has joined the Third Division club on a two-year contract, was registered as an amateur at the Irish League club.
But Cliftonville officials are adamant they should reap some financial compensation for losing one of their most promising players.
"Aidan was one of our up and coming players," said manager Laurence Stitt.
Uefa rule
O'Kane played in Cliftonville's Intertoto Cup games against Moldovan side Tiligul Tiraspol.
Cliftonville secretary John Duffy said the club would be seeking money from York.
"Aidan had been with us for four years and rose through our junior ranks.
"Uefa has a rule about clubs being compensated for developing young players and we will be taking the matter further."