The Brazil attacker, after pushing to join Tottenham over the summer, is now set to sign a new contract offered by City to 2031.
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The Daily Mail says City chiefs hope to see Savinho secured to his new deal next week.
City manager Pep Guardiola said of his young winger on Friday: "Always for the people up front it is about numbers, goals and assists.
"Savinho created a lot last season but in the final third, the final moment he was a little bit too hectic.
"But the moment he learns that, because he is so young, the more he improves, he'll be a top-class player because he can play on both sides, he's got the speed and the work ethic.
"Savinho will make this step, naturally it will come and he will be a top player for Man City."