Mundo Deportivo says Kimmich, 29, tops Barca's wish list.
The Germany midfielder, who has been playing right-back lately, has a contract with Bayern until the summer of 2025.
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Kimmich has been clear that the next coach at Bayern Munich will influence his decision about the future.
Bayern are expected to sell Kimmich this summer if he ultimately turns down an extension.
If Barcelona does not succeed in buying Kimmich, they will turn to Real Betis' Bosman prospect Guido Rodrguez.