"I hope he stays at Southampton, I really do. He took over at a difficult time but I feel he should be the man who takes them up next season," Redknapp junior told BBC Radio Five Live.
"I don't think he realised the bitterness of the feeling towards him when he left Portsmouth and joined the Saints.
"But I genuinely think he will stay and I think he'll make the decision quickly because he doesn't want it dragging on.
"If he does call it a day, I'm pretty sure that will be it for him in football management."