The full-back has failed to reach the standards expected since joining the Red Devils for a fee of £30million from Southampton in the summer of 2014.
Shaw has been hounded by injuries and the likes of Louis van Gaal and Jose Mourinho have both been critical of his attitude in training.
Reports have previously linked the defender with Arsenal, Tottenham, Chelsea and Everton but City have now emerged as candidates to take him on.
Pep Guardiola currently has a lack of options to deploy on the left side of his defence, having seen summer signing Benjamin Mendy ruled out with a long-term knee injury.
The Spaniard has been forced to call on Fabian Delph to fill in but it’s understood he wants to find a more permanent back-up for Mendy.
The report claims that United are willing to let Shaw depart for £20million in January, which could prove tempting for the Citizens as they look to ensure they have enough cover to fight on all fronts.