Koeman thinks Pochettino is happy at White Hart Lane and will pit his wits against the Argentine on Sunday when the Toffees travel to north London to face Spurs, who are unbeaten at home in the Premier League this season.
The Dutchman was Pochettino's successor as manager of Southampton and has been impressed by what he has achieved with Tottenham since taking up the reins in the summer of 2014. Koeman was initially jovial in his response but became irritated by the constant questions surrounding Barcelona and refused to be drawn on a potential return to Catalonia.
"I think he's managing a big club, he's happy, I'll ask him Sunday if he wants to move to Barcelona," Koeman said in his press conference. "In the past, they talk about (him moving to) Manchester United. I was the manager after Pochettino managed Southampton, I think he's doing a really good job in Tottenham, You ask me what he will do, I don't like these questions."
When asked about whether his future may lie at the Camp Nou, Koeman said: "Every word from my side about Barcelona is too much. I don't speak about Barcelona."
Koeman did not want to talk about a potential summer switch to Spain but was more open on the subject of Everton's tough trip to Tottenham. The former Valencia boss stressed the need to nullify Spurs offensively and identified Christian Eriksen and Dele Alli as the men who could inflict serious on the Toffees.
Tottenham were in fine fettle last Sunday when they overpowered and outclassed Stoke City 4-0. Stoke boss Mark Hughes heralded Pochettino's side's attacking prowess, but Koeman is keen to learn from the Potters' mistakes.
"That's an easy answer if you lose 4-0 but I saw the game, we will defend differently than they did," the Everton boss added. "Stoke had two big chances to score in the first half but you need to stop the strengths of Tottenham. You need to stop (Christian) Eriksen and (Dele) Alli, they pose a lot of questions but they'll have questions about Everton.
"I follow football, I follow managers, teams, how they play. In several aspects they play football in the way we like to do, they try to do high pressing, they have very good players like Eriksen and Dele Alli, offensive full-backs as do we, and they're trying to bring young players through the team."