Anne Thomson, a lawyer at the firm, said it was "undeniable" that the illness had been rife on Ventura.
“Given there were repeated outbreaks onboard over consecutive months, we hope Carnival comes back to us pretty quickly," she said.
Hundreds of holidays had been ruined, according to Ms Thomson, who added that the legal action should come as "no surprise" to the cruise company.
At the time, P&O said a "wide range of approved protocols" had been put in place, including closing on-board launderettes and preventing passengers from serving themselves at the buffet.
Ventura was also deep-cleaned and unwell passengers were asked to isolate in their rooms.
Ms Thomson said: “Let’s not forget, it’s not just the time people are confined to the cabin and are too ill to leave, but also the impact it has on the rest of their holiday, as it takes some time to recover from feeling so unwell to again fully enjoy the facilities, restaurants, and locations and days out."