Rome (CNN) -- A handful of surviving passengers of the wrecked Costa Concordia cruise ship filed a lawsuit against the cruise line on Friday, the same day the company offered each of the hundreds who'd been aboard the vessel a lump sum of 11,000 euros ($14,400).
Six individuals filed the complaint asking for a jury trial and seeking retribution from Costa Cruises, its parent company Carnival Cruise Lines and two "John Does." The suit was filed in a court in southern Florida, where Carnival is headquartered.
This is not the only legal action in this case, including in the United States. Peruvian crew member Gary Lobaton filed a lawsuit, for one, on Thursday in an Illinois court, his lawyer Monica Kelly of Ribbeck Law told CNN by e-mail. Stating that she's never met those behind Friday's lawsuit, Kelly said "we will add more (plaintiffs) in the coming week."
Marc Bern, a senior partner with the New York-based law firm Napoli Bern Ripka Shkolnik that is among those behind Friday's complaint, also told CNN that his firm represents "hundreds" of passengers who were aboard the ship. He suggested that many of them could soon join his lawsuit's six named plaintiffs.
The lawsuit filed Friday states the plaintiffs first contacted Italian consumer law group Codacons, which led to the creation of "an international legal collaboration ... to vigorously advocate for passengers' rights."
The lawsuit requests "economic and compensatory damages in excess of $10 million for the six named plaintiffs (and) punitive damages, in an amount of at least $450 million."
This per-passenger figure is far larger than the $14,400 amount decided upon following a meeting between Costa Cruises and consumer groups, which was announced earlier Friday by the Italian Association of Tour Operators.
The massive Costa Concordia liner struck rocks and rolled onto its side in shallow waters off an island on Italy's Tuscan coast on January 13, leading to a panicked overnight evacuation and at least 16 deaths. Another 16 people are missing.
Franco Gabrielli of Italy's civil protection agency, who is heading the rescue operation, said 14 of the bodies found have now been identified.
The lawsuit filed Friday faults the cruise ship operators on several counts, including "maritime negligence," "gross negligence," "intentional infliction of emotional distress" and "negligent retention."
Outlining the compensation deal it agreed to, Costa expressed "its profound condolences to the families of the victims, our continued sympathy to the families of the missing, and our deep regret and sorrow for the damages and hardship the Costa Concordia accident caused to all its guests."
The compensation would be paid to each passenger regardless of age and will cover damage to and loss of property and any psychological distress suffered, it said.
In addition, Costa said it would reimburse the cost of the cruise and additional travel expenses and will return the contents of cabin safes to their owners if possible, and it will also set up a psychological counseling program for those passengers who request it.
Separate agreements will be reached with those passengers who were injured and needed treatment at the scene and with the families of those who died, the statement added.
A spokesman for the Italian Association of Tour Operators said none of the passengers was obliged to sign the agreement but, if they do, they cannot sue Costa.
Jesus Garcia Heredia, who was on the cruise with his wife, told CNN he would not accept the payout.
"If we can reach an agreement, I am willing to agree not to sue, no problem," he said. "But not for 11,000 euros. I don't accept this."
Heredia said he has not yet been contacted by anyone in the company to talk about compensation.
"There was a lot of loss that day," he said, referring to personal belongings and the emotional toll of the disaster. "We had it really bad there."
Another passenger, Mark Plath, said he also wants to be compensated for the $6,000 worth of possessions he still has on the ship. He wants further compensation because he and his wife, Sarah, swam to shore.
"Also, I helped people quite a bit, to calm down on the boat, as well as leading them to shore and to cars awaiting above, quite a while later. My wife assisted a lady with blood all over her face (my wife is a nurse)," Plath wrote in an e-mail to CNN.
"I am not a fan of class-action lawsuits, but I think that Costa needs to take individual experiences and actions into account."
And Bern, the New York lawyer tied to Friday's lawsuit, called the compensation offer "an insult."
"This was not an incident that could possibly be covered by the ticket limitations," he told CNN's Erin Burnett. "This is an act of horrendous negligence."
At the same time, Bern said that the compensation requested in his firm's lawsuit -- $10 million for each plaintiff plus punitive damages around $450 million -- are high in part to get "the attention of the defendants."
"We do not know the nature and the full extent of the damages that these people have suffered," the lawyer added.
Roberto Corbella, president of the tour operators association, said the compensation offer "aims to give, after such a serious disgrace, a quick, concrete and adequate answer."
The 11,000 euro lump sum reflects Italian and international law, he said, with Costa likely to pay out about 3,000 euros more per passenger on top of that in refunds and travel costs.
He estimated the total cost of the offer at more than 40 million euros ($59 million), not counting separate agreements with the injured and families of those missing or killed.
Even after the incident, Corbella said cruise cancellations are only running about 10% higher than average.
Costa has said anyone who booked before January 13 for a future cruise and wishes to cancel may do so without penalty, provided they get in touch by February 7.
Meanwhile, the captain of the Costa Concordia, Francesco Schettino, is under house arrest and faces possible charges of manslaughter, shipwreck and abandoning ship.
Prosecutors on Friday questioned Ciro Ambrosio, Schettino's deputy on board the ship.
Entering the tribunal, his lawyer told reporters: "We have many arms to defend us with honor. We don't feel responsible."
Schettino has admitted to prosecutors, defense attorneys and a judge that he made a "mistake" in colliding with the rocks off shore. But he has brushed aside suggestions that he was going too fast, as prosecutors allege.
In a 126-page transcript, Schettino said he ran the ship aground to keep it from sinking and limit the tilting.
Efforts were under way Friday to open up new passages in the ship's hull so rescuers could access more areas, said Gabrielli, of Italy's civil protection agency.
Operations to remove 2,400 tons fuel from the liner's tanks will begin Saturday afternoon or Sunday, after a slight delay, he told reporters.
Weather and sea conditions are expected to worsen Saturday, leading to higher waves, Gabrielli said. While this will not prevent the removal of fuel, it could present more risk to the environment if anything goes wrong, he said.
Residents of Giglio island near the site of the shipwreck have complained of seeing white filaments in the sea, he said, but further testing is needed to confirm the origin of the substance.
CNN's Greg Botelho, Vivian Kuo, Livia Borghese, Mariano Castillo, Laura Smith-Spark and Pierre Meilhan contributed to this report.