The Championship side's striker is wanted by the Old Firm after impressing this season.
Campbell is out of contract at the end of the season which would allow Celtic and Rangers to do what they did with Moussa Dembele and Joe Aribo respectively and nab him on for a reduced amount via the cross-border compensation rules.
Stoke want the 20-year-old, who they signed from Manchester City when he was a teenager, to commit his future to them, while there is interest from elsewhere in England.
O'Neill, who is familiar with Scottish football, believes he would have a better chance of fulfilling his potential at Stoke rather than at Celtic or Rangers.
“Ty’s contract obviously expires at the end of the season," he said, "which is a difficult situation because he’s particularly affordable for the likes of Celtic and Rangers because of the cross border compensation.
“I’ve been very direct with Ty. I believe his future lies here. I think he will certainly get a lot more game time here than possibly at those two clubs. And that’s what Ty wants, he wants to play, so that’s a positive.
“Hopefully, we can get something sorted because I believe he has a big future here.
He added: “I believe he can play in any of the front three positions, as he’s shown, particularly in the Huddersfield game, and over time we can develop him into a top-class number nine which he has all the attributes to be.
“But to do that, I think he has to get game time and the best chance of all the clubs mentioned to do that is here, and he’s certainly a player I would like to keep.”