The Mein Shiff 6 ship departed from Palma de Mallorca on 20 September, embarking on a nine-day itinerary that took in a number of Greek islands.
The vessel left Crete on 27 September with 920 passengers and 666 crew onboard, and had reached the Aegean Cyclades island of Milos when 12 infections were detected from a sample test of 150 members of staff.
Six of those who tested positive have since tested negative, according to Tui.
The decision was made to continue to sail as planned to Piraeus port near Athens, according to Greece’s Civil Protection agency, where repeat rapid Covid-19 tests could be performed on those affected.
All crew who tested positive have been segregated from fellow staff and passengers in the interim.
“They are all asymptomatic,” a civil protection official told Reuters. “The vessel will most likely return to Piraeus port for repeat rapid Covid-19 tests.”
All holidaymakers had to present a negative coronavirus test before they could initially board the ship.
“Thanks to the health measures in place and the vessel cleaning rules, there is no reason of concern for visitors and crew members,” Tui said in a statement.
The Independent has contacted Tui for further comment.
Large-scale cruises were only permitted to start back up in Greece from 1 August, when it reopened six major ports.
The cruise industry has been one of the hardest hit sectors during the coronavirus pandemic, after several high-profile outbreaks on ships in March prompted various countries to warn citizens off cruising.
The UK’s Foreign Office (FCDO) is currently advising against all forms of cruising, barring river cruises.