World Health Organization chief Tedros Adhanom Ghebreyesus is accompanying Spanish officials to oversee the delicate operation.
Regional authorities have warned that it must be completed by Monday, when adverse weather conditions will force the ship to leave.
The only hantavirus type that can transmit from person to person – the Andes virus – has been confirmed among those who have tested positive, fuelling international concern.
The WHO said on Friday it had confirmed six cases out of eight suspected ones. There are no suspected cases remaining on the ship.
The MV Hondius had arrived at Tenerife early on Sunday morning from Cape Verde, where three infected people had already been evacuated to Europe earlier in the week.
It left Ushuaia, Argentina, on Apr 1 for a cruise across the Atlantic Ocean to Cape Verde.
The WHO believes the first infection occurred before the start of the expedition, followed by transmission between humans onboard the vessel.
But Argentine provincial health official Juan Petrina has said there was an “almost zero chance” the Dutch man linked to the outbreak contracted the disease in Ushuaia based on the virus’s weeks-long incubation period, among other factors.
Health authorities in several countries have been tracking passengers who had already disembarked and anyone who may have come into contact with them.