Phoenix, which has invested previously in luxury shoe group Jimmy Choo and San Francisco-based Golden Gate Capital are said to be in the final round of the auction, according to the Daily Telegraph.
A decision is expected in the next two weeks, although it is thought some of the offers for LK Bennett are lower than the£100 million to£150 million founder Linda Bennett had wanted.
LK Bennett was put up for sale at the end of last year. Bennett had tried to sell the business four years earlier but pulled the plug after poor Christmas trading meant offers fell short of the£75 million asking price.
LK Bennett has more than 70 shops across the UK and Ireland.