The Premier League side have contacted City about the 30-year-old's availability.
Palace are yet to submit a formal bid, but have enquired about the conditions involved in bringing Akanji to Selhurst Park.
If they are successful, it could open the door for Marc Guéhi to join Liverpool.
Guéhi, who has a year left on his contract and is unlikely to sign a new one, is keen on a move to Anfield.
Palace, however, are reluctant to allow another key player to leave following Eberechi Eze's move to Arsenal.
Sources have told ESPN that City value Akanji at around £15 million ($20.2m). He has two years left on his contract at the Etihad Stadium.
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Galatasaray have already contacted City about Akanji. German side Bayer Leverkusen have also been linked with a move before the transfer deadline.