Kilkenny is due to leave Elland Road when his contract expires and is being courted by a number of Championship rivals, with Burnley, Nottingham Forest, Ipswich and Bristol City all keen.
Tigers boss Nigel Pearson worked with Kilkenny during his time with the England Under-18 side before he switched his allegiance, and would be keen to join forces again.
The 25-year-old's agent, Pete Smith, confirmed that the opportunity to feature for a side with ambitions of returning to the top-flight would tempt his client.
"Hull are a club that would appeal to Neil, of course," Smith told the Hull Daily Mail.
"They're a big club that's been in the Premier League. They've touched it and felt it and that's where they want to be again.
"There's a good set-up there with very good management.
"Neil has worked with Nigel before with England Under-18s and knows his knowledge of the game is excellent."
Smith admitted that several sides have been in touch regarding Kilkenny's services and he expects to make a decision over his future in the coming days.
"Hull are one of the clubs to have been in contact but nothing has been firmed up as yet," he added.
"Neil came back to England on Monday (from international duty with Australia) and we'd hope to have something confirmed in the next week or so.
"Neil wants to get his future sorted quickly and we're in that process of talking to clubs now."