Speaking to BT Sport during Wednesday's broadcast of Bournemouth versus Manchester United, which Jose Mourinho's side won 2-0, Redknapp instructed Cook to stay with the Cherries for another few years.
"He’s learning the game, still. It’s alright saying he can move on, he goes to a Chelsea or a Manchester United or somewhere and suddenly he’s not a big player anymore. He’s a bit player," said the former Bournemouth manager.
"He’s better off learning his trade here. Playing every week being a big fish in a small pond."
Whether Manchester United are indeed interested in Cook remains to be seen - although Old Trafford is the last place those of a Leeds persuasion would want their academy graduate to end up.
Here's what they're saying about the 21-year-old - who some say grew up supporting Leeds' rivals from across the Pennines...
i can see the sense in a move to liverpool but not to united, a team populated by divas and melts. https://t.co/kUwaemVFa2— hat (@ellandread) 19 April 2018
he supports scum so it could happen. be a bad move. He should stay with Bournemouth for one more season as I still feel they could push up the table like Burnley— dean (@deancore) 19 April 2018
Good man Harry, keep him away FROM THE SCUM— David Kerr (@davidkerr2821) 19 April 2018
If Cook goes to Scum I'm seriously done— Darren (@D_Dunne1) 19 April 2018
Harry is spot on happens all the time our best young players go to a big club and don't get a game hope Lewis stays with Bournemouth and definately hope he doesn't go to Man Utd #lufc #mot #lewiscook https://t.co/9WQPyv0xnG— Neil Summers (@neilyboy1919) 19 April 2018
Yeah... i'm pretty sure they said the same thing when Bournemouth came in for him from us, he should stay and develop etc etc..... #LUFC— Youngsy (@Monkiimagic) April 19, 2018
In February, the Daily Mirror reported that Cook is a £20 million summer target for Liverpool, who are expected to lose Emre Can at the end of his contract.
It is unclear whether or not Leeds inserted a sell-on clause into the deal that took the York-born player to Dean Court in 2016.
Could Cook be off to Manchester United?