The Icelandic investor, which already owns a 29% stake in the business, is understood to have put an offer to the Moss Bros board.
The amount of the bid is unclear, however a price tag of £40 million has been attached to the menswear business, which operates the Moss, Moss Hire, Cecil Gee, Savoy Taylors Guild chains as well as Hugo Boss and Simon Carter franchises.
Moss Bros issued a profits warning at the end of last year.
Baugur has been linked to a bid for the business for more than a year and was rumoured to be hoping to build a menswear version of its multi-branded Mosaic Fashions business around Moss Bros. Earlier this year Baugur was granted two seats on the board of the menswear business. Baugur director and House of Fraser chairman Don McCarthy was installed as non-executive director and has been leading a strategic review of the group which kicked off in September.