Palace have been on the trail of the defender since the start of the transfer window and have failed with a bid after the Gunners rebuffed their initial £16m ($20.7m) offer with the north London club looking to get at least £20m for the centre-back.
According to the Daily Mail, the Eagles are keen to sign the defender this summer and have returned with a fresh offer for Chambers. They are hopeful of reaching an agreement with Arsenal this time around but are yet to match the north London club's valuation.
Chambers has attracted interest from a number of Premier League clubs after impressing on loan with Middlesbrough last season. The England international joined Arsenal from Southampton in 2014 in a deal worth £16m and the Gunners are looking to make a profit on the defender, who is currently out-of-favour under Arsene Wenger.
The 22-year-old made 36 appearances in all competitions in his debut season for Arsenal but was only involved 25 times in the following campaign. Chambers was loaned out to the Riverside outfit last season and now finds himself behind a number of players for a place at centre-back.
Laurent Koscielny, Shkodran Mustafi, Per Mertesacker, Rob Holding and Gabriel Paulista are all ahead of him in the pecking order and with Wenger shifting to a back three, Nacho Monreal and new signing Sead Kolasinac are also vying for a place in defence.
Chambers is contracted to the club until the summer of 2020, but lack of game time could see him move elsewhere. However, Arsenal are not budging from their valuation, which Palace will have to meet to sign the defender on a permanent basis.