The fashion brand told investors it has received early proposals over deals after its shares surged by about 40 per cent on Thursday amid market speculation.
The company, which is valued at just over £15m, said it received one approach from retail investor Spotlight Brands, with backing from restructuring and investment firm Gordon Brothers.
Spotlight has backed British brands including fitness-focused Sweaty Betty, while Gordon Brothers acquired the Laura Ashley brand last year following its administration.
French Connection said it has also received another potential proposal from brand investment platform Go Global Retail.
It said that discussions with both Spotlight and Go Global "remain at a very early stage".
"Accordingly, there can be no certainty that an offer will be made, nor as to the terms on which any offer might be made," the retailer said in a statement.
The fashion brand was founded by Stephen Marks in 1972 but has seen sales impacted by the turmoil on the high street in recent years, with the coronavirus pandemic adding to its woes.
In December, French Connection said it had seen "encouraging" sales after some stores were able to reopen, before the latest national lockdown came into force in January.
The takeover approaches come more than a year after French Connection called off plans to find a buyer following a year-long search.
A sale could also secure a payout for Mike Ashley, whose Frasers Group retail business owns an almost-25 per cent stake in French Connection.
Support The Yorkshire Post and become a subscriber today.
Your subscription will help us to continue to bring quality news to the people of Yorkshire. In return, you'll see fewer ads on site, get free access to our app and receive exclusive members-only offers.
So, please - if you can - pay for our work. Just £5 per month is the starting point. If you think that which we are trying to achieve is worth more, you can pay us what you think we are worth. By doing so, you will be investing in something that is becoming increasingly rare. Independent journalism that cares less about right and left and more about right and wrong. Journalism you can trust.