Palace have long held an interest in the centre-forward and the Guardian says they have tabled a €17m offer for him.
Palace boss Roy Hodgson sees Babacar as cover for Christian Benteke.
For his part, Babacar is keen on a move to England and Fiorentina are willing to sell. However, they may wait to see if Palace's offer flushes out other suitors.
West Ham and West Brom are also being linked with the Senegal international.