Palace see the Switzerland international as a potential replacement for skipper Marc Guehi, who is a target for Liverpool.
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Akanji is on the outer at City and wasn't considered for selection for defeat at home to Tottenham on Saturday.
The Sun says Palace have already made a £15m offer for Akanji and are waiting on City's response.
Akanji has won two Premier League titles during his three-year stint with City.