Two other binding offers for the whole group have been submitted by Italy's Diesel-owner OTB and US brand management company Bluestar Alliance, the source said.
Last Friday Cavalli said it had received five offers for the brand.
The company and its private equity owner Clessidra both declined to comment.
Cavalli held a board meeting on Wednesday to examine the bids, though the final decision rests with the bankruptcy court which granted creditor protection to the brand in April.
Damac is already a partner of Cavalli in a real estate project.
By Claudia Cristoferi; Editor: Stephen Jewkes