Rodri, 23, officially became a City player a day after the La Liga club revealed the 70-million-euro (£63 million) release clause in his contract had been triggered.
"Manchester City are delighted to announce the signing of Spanish midfielder Rodri," said a statement on the club's website.
The value of the release clause eclipses the reported £60 million City paid Leicester for Algeria international Riyad Mahrez last year.
Rodri said: "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.
"It's not just the titles they have won, but the way they have managed it, playing attacking football at all times. It's a style that excites me, as do the club's ambitions."