Henry is an academy product of Charlton, but his contract with the Addicks ends next summer and he will be able to leave if no new deal is signed.
Charlton will get compensation if he does leave as they developed him, but Garner is not in favour of letting him do so.
Garner wants Henry to stay at Charlton beyond the summer, emphasising the great potential the midfielder has.
The Charlton boss suggested to the club that it would be beneficial if they tied the 19 year-old down to a long-term contract.
Speaking to the South London Press about Henry, Garner said: “From my perspective, I want him to stay.
“My preference would be to give him a long-term contract and secure his future with this football club.
“I really like him as a player and as a character.
“He has got huge potential.
“My advice, my strong suggestion, would be for us to tie him down and hope that he wants to stay here and sees a pathway.
“I’d like for us to keep him at the football club for many years to come.”
Henry spent part of the last season on loan at National League side Wealdstone and this term has four appearances for Charlton.
He was a substitute in the victory against Queens Park Rangers in the first round of the EFL Cup, when he eqaulised late in the game to keep the Addicks in the tie, and started the victory over Walsall in the second round.
