According to The Telegraph, Palace are interested in signing the 21-year-old striker in the January transfer window.
Villa manager Smith has admitted that he is not sure whether the England international will be at Villa Park for the second half of the campaign.
The Villa boss has also said that Chelsea have been checking on the development of Abraham.
Smith told Sky Sports when asked about speculation on Abraham’s future: "I've no idea about the contract situation if I am honest. All we can do is work with him, we have a good relationship with Chelsea.
"Eddie Newton [Chelsea's loan technical coach] was up here over a month ago, looking and checking at his development. He's been developing with us all the time.”
Abraham has established himself as a key figure at Villa and is one of the best players in Smith’s team.
According to WhoScored, the young striker has scored 12 goals and provided one assist in 16 Championship matches for Villa so far this season.
If Chelsea recall Abraham in January, then it is going to be very hard for Villa to find a suitable replacement.