O'Donavan says that Cork City didn't even inform him about the bid, and that he heard the news from Hull City's Cork-born defender Damien Delaney.
He told the Cork Evening Echo: "It's the second time that Hull City have come in for me and this time there was a substantial offer on the table.
"I believe it was even bigger than what Reading paid for Alan Bennett and if that is true then I can't understand the club's attitude.
"I am disappointed and upset. They never came back to me to give me an answer one way or the other yesterday.
"It seems to me they wanted the deadline to come and go last night."
Southend, Southampton, Celtic and Sunderland are also reported to be interested in the former Coventry player.
The newspaper claims that, had the move gone through, it would have beaten the eircom League record, set by the aforementioned Bennett, who joined Reading for an undisclosed fee on Tuesday.
The deal would apparently have been worth €300,000 up front and instalments of €75,000 after 15 and 30 appearances for City, along with a 25% sell-on clause.
O'Donovan is out of contract when the Irish season finishes in November. Cork wouldn't be able to stop the player moving but would receive compensation of €180,000 if he joins a Championship club such as City, or more if he goes to a Premier League side, but still not as much as the reported €500,000 that was turned down yesterday.
Cork manager Damien Richardson said that O'Donavan should be "aiming a bit higher" than a club of Hull City's stature.