The centre-back departed Stamford Bridge at the end of last season after his contract expired, and is still looking for a new club.
The Daily Mail reports that Crystal Palace are hoping to convince the 33-year-old to join them ahead of top-flight rivals West Ham and Burnley.
Eagles boss Roy Hodgson wants an experienced leader in the heart of his defence next season, and time is running out with the Premier League transfer window set to close on August 8.
Burnley have been pursuing the player since May after harbouring concerns about the potential departures of James Tarkowski and Ben Gibson.
But Cahill could be swayed by the offer from Selhurst Park as it would allow him to stay in London and reunite with former England manager Hodgson.
Palace are also busy working on getting deals over the line for Real Betis midfielder Victor Camarasa, Everton’s James McCarthy and Olympiakos right-back Omar Elabdellaoui before the deadline.
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