Brown, 23, is expected to join Vincent Kompany’s side from Eintracht Frankfurt for a reported fee of €55 million following Germany’s elimination from the World Cup.
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According to Bild reporter Christian Falk, Bayern expect Brown to compete with Davies for the position rather than be a replacement.
"Bayern want Nathaniel Brown not as a replacement for Davies, but because they consider him a great Bundesliga talent for the future,” he said.
“It was a case of having to sign the German now to avoid risking a Premier League club poaching the full-back.”
Brown had previously been linked with moves to Arsenal and Chelsea, but has snubbed them both to remain in the Bundesliga.