Somerfield is demanding that John Lovering and Bob MacKenzie substantially increase their bid, which under-values the grocer.
The supermarket group's board met on Wednesday to discuss the approach.
Lovering and Mackenzie are offering just over£1 per share - valuing Somerfield at about£510 million.
There has been speculation that the pair would like to merge Kwik Save with Iceland, the frozen food chain controlled by Big Food Group.
Serial dealmaker Lovering, chairman of value fashion group Peacocks, is best known for his work with venture capitalist Permira. They bought DIY chain Homebase from Sainsbury's for£750 million. The business was subsequently sold to GUS for£900 million.