Speaking on a video call with international media organised by ESPN, which holds La Liga's broadcast rights in the United States, the former Barcelona captain reflected on his rivalry with Mourinho during their playing days.
"We're talking about a great coach," Puyol said. "I met him at Barcelona when I was coming through the academy and moving into the first team. We could already see he knew a lot about football.
"Then he built a good CV and later arrived at Madrid. We were at a very strong point, I remember a great rivalry."
Puyol drew comparisons with the Pep Guardiola-Mourinho era that defined Spanish football over a decade ago.
"In those years, aside from it being Guardiola and Mourinho, Messi and Cristiano, they were two great teams. We'll see how it goes for him. I still think Barcelona have a very good team and will be able to have a good season. With Mourinho, they can be stronger."
The 47-year-old was particularly positive about Rodri's move to Barcelona from Manchester City, describing the Spain midfielder in glowing terms.
"Rodri is very good news for Barcelona. We're talking about one of the best midfielders in the world. I had the pleasure of watching him live at the World Cup; I like him a lot. He gives a lot of balance to teams and fits perfectly with Barcelona's style of play. He's shown that with Spain and at City under Guardiola. He's going to help us enormously, it's a key position and now Barcelona's midfield is going to be even stronger."
Puyol also praised Hansi Flick's approach at the club, particularly his use of a high defensive line.
"I really like playing with an advanced defence. It requires commitment not just from the defenders, because if there's no pressure on the ball it can create huge problems for you," he said.
On Sergio Busquets' return to Barcelona, this time as a coach working with the academy, Puyol admitted he had not expected the move so soon.
"It surprised me because I thought it would take him a bit longer to get involved, but he's destined for it. He can help enormously with the academy tactically, but also with all his experience. He's a great leader and a great captain. Gaining experience at Barca Athletic could suit him very well."
Puyol used the opportunity to send a message of support to his former teammate Lionel Messi, who is going through a difficult period personally following the death of his father.
"Leo, stay strong, keep your spirits up," he said. "I've told him that privately. Everyone who loves him is with him and his family."
He went on to praise Messi's performances at this year's World Cup at the age of 39.
"What he's done at 39 seems incredible to me. He's the absolute leader, and if Argentina reached the final, it was thanks to Messi, because Argentina weren't very consistent. But in the final 30 minutes of every match Leo took on responsibility, even in the final, where Spain were superior. They had chances in the closing minutes and were dangerous," he said, before summing up Messi's legacy. "For me he's the best in history and an example for all the young players starting out."
Asked to name his favourite Clasico, Puyol pointed to Barcelona's famous 2-6 win at the Bernabeu in 2009, a result that all but secured the title in Guardiola's first season in charge.
"I've been lucky enough to play in many Clasicos, but the 2-6 is the one I remember most. It was a very tough season, Guardiola's first year with us. The first half of the season was very good, but Madrid were closing the gap on us. We were four points behind with four games left and we got an incredible win that virtually gave us the league," he said.
He also voiced support for the idea of staging La Liga fixtures outside Spain.
"It's been talked about for years that a match will be played abroad. It's complex for the teams to lose a home fixture, but it's an opportunity for fans who follow our competition to enjoy it live. If the clubs and La Liga decide to do it, I see it as a good thing."
On Lamine Yamal, Puyol said the teenager has every attribute needed to deliver more World Cups for Spain and further titles for Barcelona, but stressed he must stay focused on football.
"There's a lot of news around Lamine. I see him calm, competitive, with his feet on the ground. I see him competitive, but he does need to understand that he's won a World Cup and important titles, but he has to keep improving as a player and try to add more trophies to his collection," he said.
"He has everything. Spain are very young and have a lot of future ahead of them. Winning one World Cup is good; winning two would be incredible, and if there's a generation that can do it, it's this one. That's the direction I'd try to motivate Lamine Yamal in."