According to the Mirror, the south-London side have made an approach for the forward, whom the tabloid appear to value at £10 million.
Palace boss Alan Pardew is understood to be desperate to invest in his attack during the January window as he looks to establish his side in the top seven.
At the time of writing they sit in seventh, five points off the Champions League positions, but Pardew clearly feels more is needed in attack, with the Mirror noting that Emmanuel Adebayor and Simone Zaza are also being considered.
Palace’s hopes of signing Babacar may also depend on which other EPL sides decide to make a move for the powerful youngster.
The tabloid identify Leicester City and Southampton as further sides who are keeping tabs on the attacker, while the Express have previously reported that Liverpool are eyeing the hitman as a more mobile, cheaper upgrade on Christian Benteke.
However, any hopes any of the EPL quartet may have of signing the forward ought to be tempered by his recent comments.
Speaking to Tuttosport [via TMW], the Senegal frontman admitted that he was not looking to move in January and was hoping to remain at Fiore at least until 2018.
EXCLUSIVE: Aston Villa to cash in on key talents including Jordan Ayew as they prepare for the Championship
Partey time for Chelsea? Hiddink keen on Atletico Madrid wonderkid
Club official responds to Man City, Liverpool interest in €55m Bundesliga wonderkid
“I want to stay in Florence and cross the landmark of 10 years in purple, we’ll see. I arrived here at 15, in 2008, and at least until 2018 I want to stay, at least unless the club decides otherwise.
“I renewed my contract in summer until 2019, with an option until 2020. I have all the time and space to try and win something in what has been my only city since I left Senegal. Winning in Florence would be the best thing.”