talkSPORT understands that Premier League clubs are circling as Chelsea ponder the idea of selling Gallagher in January or at the end of the season.
Speaking to talkSPORT's Alex Crook, Pochettino hinted that he would be against the idea of selling the 23-year-old.
When asked about the situation regarding the player, the Chelsea coach said: "At the moment, Conor should be an important player who is still with us so that is the feeling now between the club and the player."
The England international will enter the final 18 months of his contract next month and there have been past links with Tottenham, West Ham and Everton.
But the midfielder showed his class yet again on Saturday after an impressive display against Sheffield United.
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The Blues won 2-0 as Gallagher was named captain in Reece James' absence.
Pochettino added: "I am really, really pleased because of his unbelievable commitment, he is one of the captains.
"He has a massive impact in the game. He is a box to box player and we are so happy with his performance."
Gallagher has been linked with a departure this season as Chelsea look to balance the books after a year of extravagant spending.
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Pochettino is keen on a striker with Ivan Toney and Victor Osimhen linked.
But he will need funds to land either player, who will not come cheap.
Chelsea managed back-to-back wins at Stamford Bridge in the Premier League for the first time in 2023 with a 2-0 victory over Sheffield United.
It has been a torrid year for the Blues, who finished 12th last season and started the day in the same position.
While they are still only in 10th, Pochettino will be hoping that Chelsea can finish the year with a bang.
Especially on Tuesday, when they take on Newcastle in the quarter-finals of the Carabao Cup.
It might be the club's best chance at silverware this season with no European football and the Premier League title surely already off the cards.
There are also league games to come against Wolves, Crystal Palace and Luton Town.