And Manuel Pellegrini's side are willing to offer the French international a massive four-year deal worth £5m a season.
Barcelona, Chelsea and City have all been heavily linked with the 26-year-old, who has 19 caps for France.
And the report claims City have stepped up their interest in Matuidi following the news this month that Porto star Fernando has renewed his contract at the Portuguese club.
Pellegrini reportedly lodged a double bid to lure Fernando and teammate Eliaquim Mangala to the Etihad Stadium on deadline day.
But City failed to match Porto's valuation and reports in Portugal now claim Fernando has committed his future to the club.
As a result, they've turned to Matuidi. The PSG midfielder is out of contract at the end of the season and Barcelona and Chelsea have also been mentioned as potential suitors.
But City are looking to steal a march on their rivals after Pellegrini was forced to employ Martin Demichelis in central midfield following injury to Fernandinho.