The Potters are in freefall and suffered defeat to Aston Villa last weekend.
But Gleghorn told The Sentinel: "I don't believe he should go now because I think Stoke will stay up by two points looking at everyone's fixtures.
"Then I think he's got some soul-searching to do in the summer to see if he can change the style of play, while the club has to decide which way it wants to go.
"Should the club sack him then? No, because I think he deserves better than that.
"But I don't think they will have to because if Tony thinks he fits their description still, then he should stay. If not, then I think he will decide to go anyway."