The 24-year-old cost City just £14 million when he signed from River Plate and his since enjoyed a meteoric rise, helping Argentina to World Cup and Copa America glory, as well as winning two Premier League titles and the Champions League at club level.
Alvarez scored 36 times in 103 appearances for the English champions, but often found himself still squeezed out of Guardiola's preferred line-up by the presence of Norwegian striker Erling Haaland.
"He has been two years here, we won everything together and his contribution has been massive," said Guardiola ahead of his side's return to action in the Community Shield against Manchester United on Saturday.
"He wanted to leave. He wanted a new challenge. Atletico Madrid is a top club in Spain...was a joy to work with him and hopefully he can find what he is looking for."
Guardiola's own future remains a source of speculation as the Catalan enters the final year of his contract at the Etihad.
The 53-year-old has won six Premier League titles, City's first Champions League, as well as four League Cups and two FA Cups during his eight seasons in Manchester.
Guardiola refused to be drawn on whether he would be interested in taking charge of the England national team at the end of the season but said he feels refreshed and excited as City aim to win the Premier League for a record-extending fifth consecutive year.
"I have to decide what I want to do with my life. If I want to continue here, I want to take a break, I want to be training national teams or not, so many things," he added on his future.
"When you arrive to the competition then my energy can rise again, so I'm fully I would say excited for the season because I see things that I like and then we'll see.
"It's a new challenge. We break the record last season (of winning four consecutive Premier League titles) and now we'll see what is the ambition of ourselves."
