"Yeremay wants to come, but it's a vital decision for him," Ilicali told the BBC, in comments that mark the clearest public confirmation yet of Hull's interest in the Canary Islander.
The Premier League club, freshly promoted and already active in the market after their €25m signing of Mendy from Rayo Vallecano, are said to be deploying all their resources to convince both player and club.
Deportivo La Coruna have so far shown no sign of softening their stance. The club's sporting director, Fernando Soriano, said last week that Deportivo's project "is not about selling" and that any departure would require Yeremay himself to request one, which he has not done. His release clause, following Deportivo's promotion back to Spain's top flight, now stands at €140m, a figure Ilicali described as "a very difficult deal" but one he believes is "at a point that seems positive."
Ilicali confirmed that Yeremay travelled to Kingston upon Hull this week for a brief visit, calling it a chance for the player to see where his future might lie should he decide to leave. The Hull owner set a timeframe of three to four days for the situation to become clearer.
Deportivo previously rejected a €30m offer plus €5m in variables from Sporting Lisbon for the winger last season, with the Portuguese club making several unsuccessful attempts to sign him.
Yeremay has not featured for Deportivo this pre-season as he continues to recover from the pubalgia that hampered his form last year, though Ilicali played down concerns over his fitness, noting the player has already returned to full training with his teammates.
"We know what player we're bringing in, he'll have a medical," Ilicali said. "If we're talking about a problem of three, four or five weeks, that's one thing.
"If it were something chronic, it would be different. If he had a serious problem, he wouldn't be coming. But it doesn't seem to be that, because he's already training."
Yeremay has not played a single minute in seven pre-season friendlies this year, one of only a handful of players in the Deportivo squad in that position, and he was left out of the matchday squad entirely for the club's opening LaLiga fixture.
For a player Deportivo have repeatedly described as untouchable, the coming days could prove decisive in determining whether he ends up staying in A Coruña or embarking on the biggest move of his career.