Although Walters will be available for Ipswich’s opening Championship fixture at Middlesbrough on Saturday, Keane agreed there was a good chance the transfer would be completed before the transfer window closes at the end of August.
Asked if he had been in discussion with Stoke and whether Walters was likely to leave, Keane said: "I’ve had loads, all summer and, yes, it’s likely.
"At this moment in time they have not made an offer that the owner feels is acceptable – and we’re delighted with that.
"I think he will be with us for Saturday and if anything it’s going to be in the next week or two.
"Obviously, the Premiership doesn’t start for another week or so and there’s usually fun and games with negotiations as you would expect, particularly with certain clubs and particularly with certain managers.
"I'd expect this one to drag on a little bit longer. Jon’s aware of it and it’s very hard to stop a player when he has the opportunity to play in the Premiership."