Perfume firm Coty, which is small but sells high-profile scents from people such as Celine Dion, Antonio Banderas, Beyoncé and Lady Gaga, made a bid for Avon on Monday.
And although the bid was rejected, the German billionaire Riemann family behind the move says they could increase the price. The Riemanns own not only Coty, but also Labelux, which owns fashion firm Jimmy Choo.
Avon turned down Coty's $10 billion offer, even though it was $2 billion above Avon's latest stock market value. The Avon management said the bid, worth $23.25 per share, was "opportunistic" and did not reflect "Avon's fundamental value."
Coty said the new company would be called "Avon-Coty," and that it would retain Avon's door-to-door sales model.
The perfume maker added that it would be prepared to raise its offer if Avon's books confirmed the management's self-confidence.
"We do not understand how your board's unwillingness to discuss our proposal can serve the best interests of Avon's shareholders," Coty chairman Bart Becht said in letter to be delivered to Avon CEO Andrea Jung on Monday.
"Coty is attempting to obtain a 'free look' at Avon on the absence of any commitment whatsoever to close a transaction at any price," Avon said in response.
Avon, with its proud 126-year history, has been struggling for some time. Although turnover increased by four percent to $11 billion in 2011, net profit sank 15 percent to just $500,000.
The US company has remained true to its unique selling point – the "Avon Ladies" who make house calls carrying a range of products. It is said to have more than six million sales reps around the world.
In contrast, Coty has only 12,000 employees, and last year's turnover of $4.1 billion was less than half of Avon's - but it is considered one of the more ambitious players in the fragrance market.
Coty is owned by the reclusive German Reimann family, valued by financial monthly Manager Magazin at €8 billion ($10.7 billion). The Reimann family lives close to the western German city of Mannheim.
Avon says it has 3.5 million customers in Germany.