The French champions are in negotiations to extend the contracts of their two stars, whose deals expire in June 2022.
Mbappe has been a reported target of Real Madrid in recent seasons while Neymar has been linked with a return to Barcelona ever since he left the Catalan club for a world record €222 million transfer fee in 2017.
"I believe the club is working on it," Pochettino told Cadena Ser. "They are obviously making an effort to renew the contracts of the players they believe are important not just for the present but for the future."
Earlier this week, PSG sporting director Leonardo said that he has "good vibes on both those cases."
Asked if he is worried about losing either Neymar or Mbappe next summer, Pochettino said: "You always want the best players at your club. Paris [PSG] has the strength and capacity to convince and prove how important these types of players are. I have no doubt whatsoever that the club will do everything possible to keep the players."
Pochettino is not surprised that many clubs are monitoring Mbappe's contract situation.
"It's clear that Kylian is a player with enormous potential," the Argentinian, who took over PSG earlier this month, said. "He's 22 and no one is going to now discover Mbappe. Which coach wouldn't like to have a player like Mbappe in their team for a long time!"
"The other day, he had bit of frustration for not scoring but it's normal," Pochettino added. "Sometimes things are taken out of context.
"Kylian is very happy here in Paris. We have a very good rapport. We will see what happens. But I have no doubt that Kylian loves to be here in Paris."
Meanwhile, Pochettino refused to be drawn on transfer speculation linking Lionel Messi to PSG this summer after Leonardo said the club are following his future "closely."
The Barca and Argentina captain becomes a free agent in June and has not signed a contract extension.
Asked if he was surprised that his compatriot Messi had not signed a new contract with his all-time club, Pochettino said: "You have to respect the decisions made by the player and by the club [Barca].
"There are reasons for that and we, from the outside, only see a tiny part of what is happening on the inside. The most intelligent thing is not to comment."
Pochettino remains in quarantine after testing positive for COVID-19.