Norwich have already failed with two bids for Walters, who handed in a transfer request on the eve of deadline day.
And it is understood the Canaries have competition from a third Premier League club, believed to be West Brom, as well as Leicester, who have now made an approach.
None of the interested clubs has yet matched Stoke’s £3million valuation of the Republic of Ireland international, and the club are believed to be reluctant to let him go without signing a replacement.
That could yet happen, as it is understood Stoke are in negotiations over a potential deadline-day deal.
Mark Hughes said at the weekend he would prefer to keep Walters, who is in the final year of his Stoke contract.
“He knows my feelings – we want him to stay,” the manager said. “Norwich did make a bid but it was nowhere near our valuation of what Jon gives us, so that was sent back very, very quickly.”