The 22-year-old signed a five-and-a-half-year contract with the Club in January but remained at River Plate on loan until the end of the South American campaign.
His final match for River came on Thursday in the second leg of their Copa Libertadores last-16 match against Velez Sarsfield, a game which finished 0-0 and saw Velez progress 1-0 on aggregate.
“I am incredibly happy and I feel ready to play for City,” he said.
“This is, without doubt, one of the biggest teams in world football. You only have to see what this team has won in the last decade to understand the quality of the setup.
“I feel confident I can flourish here. The style of play Pep encourages is exciting and I can’t wait to be a part of it.
“City has an amazing recent history of Argentinian players and I want to put my stamp on the Club in the same way Sergio Aguero, Pablo Zabaleta and Carlos Tevez did.
“I am so thankful for everything River Plate did for me and I want to thank the fans, staff and players for their support.
“But I am now 100% focused on Manchester City. I want to give my best to this club, help my teammates and contribute to Manchester City’s success.”
Txiki Begiristain added: “I said when we announced the signing we were getting a player of significant potential and everything we have seen from him since then has only confirmed that.
“He scores all kinds of goals and has real variety in his game.
“We feel with the manager and staff we have here, he can develop even further and become a really top player.”
Alvarez made his senior River Plate debut in October 2018 and has since forged a reputation as one of the best young forwards in Argentina.
He managed 53 goals and 31 assists in 120 River Plate appearances – including a strike in their 4-0 win over Racing in November, a result that sealed the Argentine league title.
Alvarez made his senior international debut in a World Cup qualifier against Chile in June last year, replacing Angel Di Maria with an hour gone, and was part of the squad that won the Copa America in July.
In total, he has nine caps for Argentina, scoring once.
Everyone at Manchester City is looking forward to watching Julian and we wish him all the very best during his time here.