Germany international Sane endured a frustrating 2018-19 campaign at the Etihad Stadium on an individual level, falling down the pecking order in Pep Guardiola's squad since the turn of the year.
However, Raheem Sterling and Bernando Silva were granted a more prominent role ahead of Sane and as a result, there have been claims he could seek regular football elsewhere while the transfer window is open.
Bayern Munich have confirmed their interest in the former Schalke attacker, but no formal offers have yet been submitted amid concerns over City's lofty €100m ($113m, £89m) valuation.
Kimmich, who is currently working with Sane as part of Joachim Low's Germany squad ahead of upcoming Euro 2020 qualifiers, has welcomed talk of Sane's arrival at Bayern.
The right-back feels his national international team-mate has not been able to reach his full potential at City due to a lack of opportunities, as he told Bild: "I would like to have Leroy at Bayern, that would send the right message.
“He is a player who would be a good fit at Bayern. I think he is a very good player.
"He has great potential but was not allowed to play at City last season, which I cannot understand.
“But it is his decision and the task of those responsible to convince him.
“If I was in the club’s position, I would certainly buy him.”
Arjen Robben and Franck Ribery left Bayern at the end of last season, which has left head coach Niko Kovac short on options in wide areas.
Sane could certainly help soften the blow of a double departure, but City are reluctant to sell a prized asset and hope to tie him down to an extension at the Etihad instead.
Bayern have already completed the signings of Lucas Hernandez and Benjamin Pavard, with a number of additional reinforcements expected to arrive during the summer market.