The England Under-21 international goalkeeper is currently on loan at the Tigers from Fulham, where he has found first-team opportunities hard to come by.
With Hull currently occupying the second Championship automatic promotion spot, they could be in a position to hand Stockdale his top-flight chance next season if they are prepared to match Fulham's £2.5m valuation.
"Staying at Hull permanently is a possibility. I have to think about it, especially if they go up," The Mirror quotes him as saying.
"It has been very frustrating at Fulham. There have been bits here and there, but that is not down to me - it's because they have a keeper in Mark Schwarzer who is doing very well.
"But there is a fantastic group of lads at the KC Stadium, and the team also plays great football."
Stoke City goalkeeper Asmir Begovic has been linked with a move to Fulham, an arrival that would bring Stockdale's future at Craven Cottage into question.