Clyne, 23, is being targeted by Liverpool while Pochettino is on the verge of joining Spurs.
Despite bigger clubs awaiting the pair, the defender wants to continue his alliance with the Argentine tactician, who he claims has improved his game.
"He has been so different to any other manager," Clyne said.
"He planted it into our heads that even if we were up against bigger teams and bigger players, we could compete at the same level as them.
"He taught us never to fear them or be intimidated just because of their big names or their big reputations.
"He'd have us pressing high, keeping a high line, receiving the ball in difficult situations, keeping possession and basically having the confidence to play football rather than being afraid.
"The understanding in our defence was down to our training. We worked really hard on our fitness and our intensity and it showed in the games.
"For me personally it took my game to another level. It would be a real shame if we were to lose him."
Clyne, who plied his trade with Crystal Palace for four seasons before switching to the Saints in 2012, was in the mix to be selected in Roy Hodgson's England squad to go to Brazil, but the modest London native said his time had not yet come.
"I would have loved to have been called up," Clyne said.
"But I still feel that the defenders in my position that are better than me at the moment. So I need to improve to get to their level.
"I still feel my time will come."
