Sky Sports says City see the Swede as a readymade replacement for Erling Haaland.
City haves already secured Hugo Viana as new sporting director for next season. But it is far from certain that he will be the only recruitment from Sporting.
It is uncertain that Haaland, 24, will extend his contract as he is attracted by a move to Real Madrid.
He is said to have a buyout clause of €150m. City have already started looking for a replacement.
And Gyökeres, 26, is at the top of the wish list, if Haaland chooses to move on in his career.
The Sweden striker has been a huge success since moving to Sporting in the summer of 2023 from Coventry City.
Gyökeres has a contract until 2028 and has a buyout clause of €100m.
