Right-back Kimmich, 24, is increasingly at odds with his bosses at the Allianz Arena over which position he sees himself eventually playing in, as he wants to feature as a central midfielder entering his peak years.
It was under Pep Guardiola that Kimmich got his big break and the City boss has continued to keenly follow his impressive career progression.
City will lose David Silva next year and possibly Fernandinho as well, so might require another body to supplement the youthful legs of Rodri and Phil Foden.
The Citizens are demanding £138m for Sane, who is out until the new year after he damaged cruciate knee ligaments playing against Liverpool in the Community Shield win earlier this month.
The belief inside the Etihad is Bayern simply do not have the cash and have instead tried to force his price down with their public pursuit of the player all summer.
City's new contract offer to Sane remains on the table but if Bayern do come calling again Kimmich, who is valued at around £65m, could have his name cheekily thrown into the pot by the Premier League champs to counter any low-ball offer.
Kimmich has three years remaining on his Bayern deal but his club are planning a major squad rebuild and the tough-tackling star does not want to be left behind on wages.
The Citizens would certainly be able to better his salary at Bayern and move to the Premier League might appeal to the man capped 42 times by Germany.
Plenty of German stars have shone in England's top division and the Guardiola connection means that it is not fanciful that the versatile star might end up playing for the blue half of Manchester.
The boss has an excellent record of recruiting the right players and, although keen to keep Sane, bringing in Kimmich might be of benefit in the long run.
Whether the Bundesliga giants are willing to do the deal is open to question but their pursuit of Sane is expected to continue and it may be that compromise is needed from both parties for the saga to have a happy ending.
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