The new offer made late on Monday, ends several months of suspense over Bonomi's intentions.
He is offering significantly better terms than the Fosun-Ardian bid which is backed by the club's management.
Bonomi is offering €21.0 per share compared with €17.50, and €22.41 per share for each bond which may be converted into shares, compared with €19.79.
Trading in shares in Club Med had been suspended on Monday after the stock had risen by 2.74 percent to €19.51. When trading was resumed early in the afternoon on Tuesday, the shares gained 10.10 percent to €21.48, slightly above the price in the latest offer.
Analysts at brokers Gilbert Dupont said commented that Fosun-Ardian might raise the terms of their offer, but said that the terms offered by Bonomi appeared "attractive".
The new bid was put formally before the French stock market authority AMF by Global Resorts SAS which is owned by Bonomi's company Investindustrial.
Fosun and Ardian launched their offer a year ago but it ran into opposition from some shareholders who said it was pitched too low, since the shares were then worth about €19.0.
That first offer valued the company at about €550 million.
Bonomi gradually built up an interest of ten-percent in the Club Mediterranee, and then made his move overnight.
The company Gaillon Invest, which is the vehicle for Fosun and Ardian, said that it had taken note of the counter offer which it would study once it had access to the official proposal.
An investment fund which owns shares in the club, called CIAM, and which had made a legal challenge to the Fosun-Ardian bid, welcomed the move by Bonomi.