Rick Reed's son said the couple were in "good spirits and recovering as best anyone can", given the situation.
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"There is a possibility, that in next couple weeks, Rick and Ivy will be flown back to a hospital in America, to continue their treatment," he said.
"This is dependent on their health status remaining stable. When this will happen is still unknown, so everyone needs to remain patient and keep thinking positively.
"We are just taking everything one day at a time. We can't thank everyone enough for the overwhelming love and support."
There were 47 people on the island at the time of the eruption, which has killed 15 people according to the New Zealand Police.
Another 15 people remain in hospital across New Zealand, while a further 13 patients have been transferred to hospitals in Australia.
Two people remain unaccounted for on or in the vicinity of Whakaari/White Island.
Yesterday, a Police National Dive Squad of nine members searched the waters around Whakaari/White Island for a missing body that had been seen in the water after the eruption.
However, "unique and challenging" weather conditions hindered the water search and no additional bodies to the six recovered on Friday was made.
Police Deputy Commissioner John Tims said conditions in the water around White Island yesterday were "not optimal", with between zero and two metres visibility.
"The water around the island is contaminated, requiring the divers to take extra precautions to ensure their safety, including using specialist protective equipment," Tims said.
"Divers have reported seeing a number of dead fish and eels washed ashore and floating in the water.
"Each time they surface, the divers are decontaminated using fresh water."