The 26-year-old arrived at St Mary's from Plymouth having played for no less than seven clubs and he failed to immediately settle, leading to further loan spells with Millwall, Blackpool and QPR.
Despite criticising club chairman Nicola Cortese on his return to Saints in January 2012, Puncheon gradually became a first-team regular and has impressed in the Premier League this season.
With Puncheon's contract up in the summer, Pochettino said through an interpreter: "I myself haven't spoken anything about his contract.
"I've only recently arrived so my first priority has to be the whole of the team, not just one player. There are certain movements that were already happening with players when I arrived.
"In the coming weeks we will be speaking at a club level with the appropriate people to decide on the matters of players at the club.
"Puncheon is a player I am counting on and he is a player that has certain qualities and skills that are good for the team so, yes, I'd like him to stay."
He added: "I always think that in football it is not just one person signing any players. It is the club signing the players.
"Now I am the manager here and I will probably bring in some players.
"Some players will work out so maybe sometime in the future I may be thanked, but then there will be players that didn't work out. That is how football really is."
One of the clubs linked with a move for Puncheon is Newcastle, who host Southampton on Sunday, and Pochettino added: "It will be down to the player to make that decision. I've said before that we will make decisions on the future."