He outlined a strategy of investing more aggressively to take advantage of tourism growth in Europe and Latin America, adding six new holiday resorts and raising the number of beds to 20,000 worldwide from 15,000. He also proposed investing another $205 million in the resort group than its current management is envisioning.
Bonomi said there were numerous ways to modernize the group and to make its management more transparent and accountable, both of which, he said, would help to improve profitability.
Club Med's current management plan is focused more on Asia than is healthy, Bonomi said, because "no enterprise should bet its whole future on China, on one country."
"The problem with Club Med is a lack of vision," he added.
Club Med said in a statement that it would study the new offer. A spokeswoman, Caroline Bruel, declined to comment further.
Ardian did not immediately respond to requests for comment, and Fosun could not be reached for comment. But Ardian had said previously that the two did not intend to raise their offer.
Bonomi's investment vehicles own nine-tenths of Global Resorts, the consortium behind the new offer. The remainder is owned by Sol Kerzner, a South African hotelier; the PortAventura amusement park group; and GP Investments, a Latin American private equity firm.
The new bid threatens to further delay negotiations that have been drawn out for more than a year amid complaints from minority investors. It also runs the risk of running afoul of French politics. President François Hollande's government has taken an activist approach toward foreign investments in businesses that are considered national champions, and there can be no more of a French holiday company than Club Med.
Fosun and Ardian had already won the backing of Club Med's board. The two companies have said that they want to redirect the group's center of gravity toward Chinese vacationers as the European market stagnates, even as they maintain the decision-making center of the operation securely in France.
Bonomi, a former Lazard banker, is the heir to an Italian industrial and real estate fortune. His investment vehicles already own more than 10 percent of Club Med's shares, making him the largest single shareholder, with slightly more that Fosun's 9.96 percent and Ardian's 9.4 percent holdings.
Henri Giscard d'Estaing, Club Med's chief executive, said last year that, working with Ardian and Fosun, the goal was to attract 200,000 Chinese guests next year, more than double the level of 2013. Club Med opened its third Chinese resort in June, on Dong'ao Island, near Macau.
In May, the French market regulator, the Autorité des Marchés Financiers, issued what is known as a put-up-or-shut-up order, giving Bonomi until Monday evening to either make an unconditional offer or drop his bid for at least six months. It also agreed to delay the closing of the Fosun-Ardian deal, giving Bonomi time to examine Club Med's books and decide on his move.
Club Med operates 70 resort "villages" in 40 countries, offering packaged family holidays for a relatively well-off customer base.