The 26-year-old joined from Plymouth just over three years ago but his time at St Mary's has not always been plain sailing.
Puncheon spent several loan spells away from the club after losing his place in the first team and little over a year ago took to Twitter to criticise Southampton executive chairman Nicola Cortese.
The winger threatened to see out the rest of his contract training with the reserves and youth team, but the pair patched up their differences and he has now become an integral part of the team.
So much so, in fact, that Southampton fans are clamouring for Puncheon to extend his stay beyond the end of the season - something Pochettino would also like to see.
"I myself haven't spoken anything about his contract," the Argentinian said, speaking through a translator.
"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."