"Leonardo called me and said: 'The possibility is there. Would you like it or not?' The good thing is that he called me to ask. I said to myself: 'Is it a question?' I thought it was a joke, no? Of course," Pochettino said.
"I told him: 'Do we have to go and get him? Do I have to drive the van?' And that is when the negotiation started. From then on Leonardo would call me every night to tell me how the situation was."
Messi, 34, had agreed a new five-year contract with Barca, with his previous deal having expired on June 30. However, that contract would have meant Barca's wage bill would stand at 110% of their revenue.
"Everyone had assumed that Messi would continue at Barcelona," Pochettino said.
"Many clubs, if not all clubs, dreamt of having Messi, with him being a free agent. From the moment that PSG showed an interest, Leo wanted to come here. Everything happened very fast. We must highlight the work done by Leonardo, by our president and all the club executives that in two or three days were able to complete the signing of the best player in the world."
Leonardo, PSG president Nasser Al-Khelaifi and club executives, meanwhile, are still working on convincing Kylian Mbappe to remain at the club beyond next summer when his contract expires.
"Who can't think that PSG wouldn't want him [Mbappe] to continue here for many more years?" Pochettino added.
"The club will try to convince him and seduce him to continue here."
A long-term target of Real Madrid, Mbappe confirmed that he had wanted to leave PSG in July, but the Ligue 1 giants refused to sell him despite knowing the France striker could sign for a new club on Jan. 1, as he will be a free agent next summer.
Mbappe's mother, Fayza Lamari, recently said her son is in talks to sign a new deal with PSG despite intense interest from Madrid.
"He is clear about things and loves football," Pochettino said. "He is 22 but is very mature to manage this type of situations regarding his future. I have no doubt in my mind that Kylian can face any challenge that the future brings him.
"You cannot underestimate football players. They have their entourage, their advisors, but it is them that take their own decisions. Kylian will take the decision he needs to take, and the club will do everything possible to keep Kylian because we are talking about one of the best players in the world. He is 22 and has great potential."
Neymar, 29, said Qatar 2022 could be his final appearance at the tournament because he was not sure if had the "strength of mind to deal with football anymore".
"He is a sincere man, he expresses his feelings quickly, but his mental strength, in a life that has been under the spotlight for a long time, is great. He has suffered a lot pressure since he was little," added Pochettino.
"Sometimes things get taken out of context. Neymar loves football and I'm sure he will play for more years. I have no doubts."
Pochettino confirmed that centre back Sergio Ramos was still unavailable due to injury and will go undergo "individualised training" for another 10 days as he looks to play his first game for the club after arriving from Real Madrid.
"I have no doubt that Ramos will be able to find his best level. Sergio's situation is not pleasant because he has not had options to train with his team mates or to play for a long time. He is suffering, but he is strong and everyone at the club is helping him," he said.