The Spanish midfielder joined the club on Thursday from Atletico Madrid after paying a 70 million euro buyout clause on Wednesday, reports Efe news.
"What City have achieved in the last two years has been amazing and I'm looking forward to being part of such a talented squad," the 23-year-old said.
"It is not just the titles they have won, but the way they have managed it, playing attacking football at all times. It is a style that excites me, as do the Club's ambitions," he added.
Rodri, who had a five-year contract with Atletico, is City's coach Pep Guardiola's second signing this summer after Spanish defender Angeliño, who has joined from PSV Eindhoven.
"I cannot wait to start working with Pep Guardiola and my new team-mates and hopefully we can achieve great things together," Rodri said.
He became the fifth player to leave Atletico ahead of the upcoming season after Lucas Hernandez's transfer to Bayern Munich was announced in March and Diego Godin and Juanfran Torres's contracts ended on June 30.
Antoine Griezmann, the 2018 FIFA World Cup winner with France, has also announced that he is to leave the Madrid side, paying his 120 million euro clause this month and has reportedly been linked to a move to Barcelona.
Rodri, who has been raised in Villarreal's youth teams and joined the capital side last summer, played a total of 47 matches out of 51 with Atletico across all competitions in the 2018/2019 season.
Only Griezmann and Angel Correa played more games for Atleti last season, with 49 and 48 matches, respectively.
( With inputs from IANS )