Bonomi has been locked in a dragged-out fight with Chinese conglomerate Fosun and its partners over Club Med, which is coveted for its potential to appeal to new middle classes in emerging economies.
In a statement, the AMF announced that Bonomi is now offering €23 a share, which values the company at €874 million.
The offer is bigger than the Fosun bid - worth €839 million - which the board of directors of the French holiday firm unanimously recommended last month.
It is also much larger than an offer previously made by Bonomi.
According to a source close to the situation, US private equity giant KKR is in talks to be a small co-investor alongside Bonomi.
Club Med first became a high-profile name in the European tourism industry because it offered holidays in hippy style villages where sports activities were included.
It has since moved up-market, weathering financial storms in the process, and is now looking for further expansion, including in China where it has been a partner with Fosun.