Leicester manager Ranieri was keen on taking Argentina defender Pochettino to the Vicente Calderon Stadium when he was manager between 1999 and 2000.
The pair meet for the second time in four days when the Foxes travel to Tottenham this evening.
Spurs boss Pochettino was at Atletico’s La Liga rivals Espanyol when Ranieri moved in only for money issues to halt the switch.
He said: “Yes I did (try to sign him) but not now!
“It was probably when I was the manager at Atletico Madrid, I always liked him as a player.
“I was very, very close but there was a little problem with our money at Atletico Madrid, so it wasn’t possible.
“He was a player I always admired, a very good player.”
Pochettino insists he had a lucky escape when Ranieri’s attempt to sign him fell through.
“Claudio tried to sign me at Atletico. We spoke a little bit about that on the touchline on Sunday for two or three minutes before the start of the game,” Pochettino said.
“He said I was lucky because it was the year Atletico went down!
“The deal was done and I had agreed to go but there wasn’t the money — it was impossible to pay because there was a takeover at the club.
“It was lucky.”
The teams drew 2-2 in the FA Cup on Sunday and Spurs are fourth, four points behind second placed Leicester as they both chase a Champions League spot in the Premier League.
Jamie Vardy is set to make his Leicester comeback after returning to full training.
The England striker joined in with the Foxes squad training yesterday ahead of the trip to Tottenham.
Vardy missed Sunday’s 2-2 FA Cup draw at White Hart Lane as he was recovering from a minor groin operation earlier this month.
He had done some light training away from the squad on Monday but has now rejoined the group and the Foxes expect to have him available to face Spurs.
Claudio Ranieri had planned to assess his 15-goal top scorer yesterday and believed Vardy, who has not netted in his last four games, would be ready.
He said: “Jamie had a problem for a month, more or less, and he continued to play. Now without the problem he is much better.
“I’ll see if he starts from the beginning or if he is ready to come with me on the bench.
“Every time he wants to play, I know him.”
Meanwhile, Spurs boss Pochettino believes Dele Alli fully deserves the new contract that will keep him at White Hart Lane until 2021.
Alli has been among Spurs’ best performers this season since completing a £5m (€6.6m) move from MK Dons, and he is the Barclays Premier League’s leading scorer from midfield, with five goals in 18 appearances.
The 19-year-old’s outstanding form has resulted in selection by England in November, just three months after making his debut in the top flight.
“It’s very good news for us and for him,” Pochettino said. “After the last six months, Dele deserves to improve his salary and his contract.
“He has achieved the level of the team-mates and now he is international and plays regularly in first team. He deserves all he takes now.
“All the players are important. Dele is important too, but for me the most important thing is the collective, the squad, the spirit.
“When you have a very good player with enough quality to play, you can achieve what you want.
“The most important thing is that players like Dele are here and they show commitment with the club.”