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Spanish newspaper Marca reported that City owner Sheikh Mansour made the offer, which also contained a £44m-a-year salary.
Real Madrid's Sergio Ramos, Casemiro and Dani Carvajal all remain unavailable for Saturday's La Liga game at home to Sevilla.
Cristiano Ronaldo's goal output has suggested he needs to go back to the training ground and work on the basics until he becomes superhuman again.
In May, Madrid became the first team to retain the Champions League in its current format and won the Liga trophy for the first time in five seasons.
Gremio's Arthur has confirmed he is in contact with Barcelona but denied there is already an agreement in place for him to move to the Camp Nou.
Ronaldo, 32, helped Madrid to a La Liga and Champions League double last year and is favourite to win a fifth Ballon d'Or on Thursday.