The former Manchester United man has been linked with a move to Chelsea and Barcelona but City could steal a march on their rivals with a bumper offer.
The Mirror claim Barca can only afford to spend £57m, which has paved the way for a City swoop.
And the Juventus star could arrive in Manchester as early as next week as Manuel Pellegrini is keen to do a deal before their two-week tour to Australia.
Neighbours United would get a £1.25m windfall under FIFA rules as the Frenchman spent three years at Old Trafford.
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Manchester City could break the British record transfer fee to sign Paul Pogba
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There were concerns that City could breach Financial Fair Play regulations, but sanctions have been lifted and the 2014 Premier League champions are now free to pursue a deal.
They are believed to be the only club who can match the Turin outfit's asking price, as well as the 22-year-old's demands of a contract worth £240,000 a week.
If City complete a deal, the fee would surpass the biggest fee paid by a British club by some distance.
United are the current holders of that record after bringing Angel di Maria to Old Trafford from real madrid for £59.7m last summer.