Seagulls boss Chris Hughton targeted the former Denmark Under-21 international and was seen to be in talks with Cardiff owner Vincent Tan after watching Zohore play against Sheffield United on August 15.
If the two clubs had agreed on a deal it would have meant Hughton would have broken the club's transfer record for the third time in a fortnight after the £13.5m swoop for Jose Izquierdo, which itself came just days after Brighton paid PSV Eindhoven £10m for midfielder Davy Propper.
"It would have been a record purchase for Brighton and a record sale for Cardiff," agent Darryl Powell from Paramount Sports Management told BT.
"It would also have been a record for the sale of any Danish player.
"It was a real, really honorable amount that was offered. And there was also strong interest from Everton."
However, Cardiff stuck to their guns and rejected the offer, knowing that Zohore would not kick up a stink because of his respect for manager Neil Warnock.
"There were several factors that meant that the deal did not turn into anything," added Powell.
"I think it was because Cardiff are doing very well and were at the top of the Championship, and Kenneth is very proud of everything Cardiff manager Neil Warnock wants.
"The manager sees a great star in Kenneth, and so I think the board and owners in Cardiff are really happy with Kenneth.
"But there is also an important party in this, and it is Kenneth, who is self-satisfied wherever he is. He has an incredibly mature approach to it and respects his manager really much."