The pair offered £510m in April for the Bristol-based company which Somerfield turned down as it “undervalued” the group.
Somerfield said this latest offer undervalued the company and posed “considerable risks”.
Chairman John von Spreckelsen said: "We have strong brands in Somerfield and Kwik Save, a solid strategy for delivering shareholder value and excellent prospects.
"The strategy is underpinned by a very strong balance sheet and a valuable property portfolio."
Meanwhile, Somerfield has revealed that its renewal programme for its Kwik Save stores and the introduction of its new own label value range Simply has attracted new spending and gained market share from rivals.
The chain has seen £2.2m in new spending with 76% of this spent on Kwik Save’s Simply brand.
Gill McComas, Kwik Save marketing director, said: “The fact Simply is bringing us new customers from our rivals is the icing on the cake."