The 20-year-old has been a big player for the Basque side so far this term, but reports suggest that offers of above £30m may be accepted in January.
This has lured City in, despite the addition of Eliaquim Mangala over the summer.
Manuel Pellegrini is understood to not yet be content with his current options, with Vincent Kompany still without a top-class partner.
Laporte is seen as a good long-term option to play alongside the Belgian, who is widely hailed as one of the best players in his position in world football.
United are also interested, but cannot offer Champions League football.
Chelsea could also move, yet Jose Mourinho may look to further assess summer arrival Kurt Zouma before committing.