Messi, 31, was still in the early stages of his senior career at the time, but ended up staying at Camp Nou and developing into one of the best players of all-time.
Pochettino – then in the latter stages of his playing career at Espanyol – said Messi almost joined the club after impressing against Juventus in the Joan Gamper Trophy.
"I started to hear about him when I was at Espanyol – a small guy playing in the academy in Barcelona," Pochettino said ahead of his team's clash with Barca in the Champions League on Wednesday.
"He arrived when he was 13 years old, going from Argentina to Barcelona. Then I heard he was so close to signing for Espanyol – very close – in the season Barcelona played against Juventus under [Fabio] Capello in the Joan Gamper Cup.
"Because he was so fantastic in the Joan Gamper, Capello praised him afterwards and they changed the decision and kept Messi at Barcelona.
"So if you look back in history, maybe if he didn't play in the Joan Gamper that summer, maybe today he could have been the biggest star in Espanyol and Espanyol could be Barcelona.
"We would have been team-mates, and maybe I would have still been in Espanyol managing him."
Messi made his league debut for Barca in October 2004, facing Pochettino and Espanyol in a 1-0 win.
However, Pochettino said it was tough to remember that encounter, with Messi just another product of the Barca academy at the time.
"When Messi made his debut, it was like another youngster that had arrived to the Barcelona team from the academy – like it was Xavi or [Andres] Iniesta or Ivan de la Pena," he said.
"Actually, no. When De la Pena started to play for Barcelona [in 1995], there was an explosion of excitement.
"I always remember Ronaldo saying he was the best player that he saw in his life and he played with."