MSC denied “incorrect” media reports, but would not say how many cases had been picked up during routine testing.
“Today we are disembarking 45 positive cases in Genoa,” it said in a statement.
A spokesman later told AFP that an unspecified number of other passengers who tested positive were still onboard and would be let off at future stops.
The company said most were asymptomatic and they and their close contacts were immediately isolated in cabins with balconies and provided with medical attention.
Port authorities in Genoa confirmed to AFP an unspecified number of passengers had been disembarked, with Italians from the north of the country to be taken to their homes and foreigners transferred to care facilities while they isolate.
They said the cruise ship would continue its journey to the Italian port of Civitavecchia where positive patients living in central or southern Italy would be disembarked and taken home to isolate.
All transfers and stays in Covid facilities were being paid for by MSC Cruises, the authorities said.
The cruise industry worldwide is struggling to get back on its feet after voyages were halted at the start of the pandemic and several vessels were hit by outbreaks.