The 23-year-old has been exceptional during Chelsea's four-match unbeaten run but his current contract expires at the end of next season.
Chelsea have been open to selling the homegrown star, with Everton seeing a verbal £45million bid accepted in January, while both West Ham and Tottenham weighed up moves over the summer.
But Gallagher stayed put and has since gone on to prove his worth under Pochettino. Asked whether he wants Gallagher to sign a new contract, the manager said: "It is like a marriage between the player and club, both sides need to feel good and that is why this moment you can arrive at a new deal.
"Even if you want to extend the contract, you stop and speak. Of course, he is a player that is doing fantastic. We are so happy with how he is doing.”
Gallagher is expected to continue in midfield alongside Moises Caicedo and Enzo Fernandez at home to Brentford tomorrow.
The trio impressed together with the likes of Lesley Ugochukwu, Romeo Lavia and Carney Chukwuemeka all spending time on the sidelines injured.
Pochettino will likely stick with the team that drew 2-2 against Arsenal last weekend having seen his side outplay Mikel Arteta's title-chasers for the majority of the match.
The Blues were unfortunate not to claim an important win at the Emirates as they saw a two-goal lead wiped out in part due to a mistake from goalkeeper Robert Sanchez.
The goalkeeper booted a clearance straight to Declan Rice who empatically finished from distance before Leandro Trossard levelled minutes later.
Asked how Sanchez reacted to criticism following his mistake, Pochettino added: "He is very relaxed, he is very calm. He is very good. We are there to try and help. We are helping him. He is calm. He knows, we know, football is about mistakes.
"The responsibility is with the whole team. The coaching staff is the first as we transmit the idea of how we want to play. To blame someone is to blame us, myself, but not him. He is delivering what we want and of course, maybe you can miss a penalty, you can miss a pass.
"He is showing great character and personality. He is going to be good tomorrow. I have no doubt about that."