The Switzerland international is seen as a potential replacement for Palace's captain, Marc Guehi, who is reportedly being targeted by Liverpool.
Akanji has struggled to secure regular playing time at City and was not included in the squad for their recent match against Tottenham.
According to reports, Palace has submitted a £15m bid for Akanji and is now waiting for a response from City.
Akanji has been a part of City's team for three years and has won two Premier League titles during his time with the club.