The offer of 30 pence share from a group including Apinder Singh Ghura, Amarjit Singh Grewal and KJR Brothers Ltd values the once dominant player in British fashion at about 29 million pounds ($39.6 million), according to Reuters calculations.
The offer price represents a 30% premium to French Connection's Wednesday closing price.
Ghura is the company's second-largest shareholder with a 24.95% stake, according to Refinitiv data, and with a 41.5% stake, founder and CEO Stephen Marks remains its largest shareholder.
Talks with the consortium are ongoing, French Connection said. It began seeking new suitors earlier this year after investment firms Spotlight Brands and Gordon Brothers pulled out of early talks to buy the company.
French Connection, once known for its provocative "FCUK" brand of clothing, has not been profitable in a decade and the COVID-19 pandemic has only hit sales harder.
($1 = 0.7315 pounds)
(Reporting by Sachin Ravikumar in Bengaluru; Editing by Amy Caren Daniel)