He was on the island with his friends Richard Elzer and Karla Matthews, who are believed also to have died but their bodies are among those yet to be retrieved.
Six deaths were confirmed after Monday's eruption.
The family has not yet issued public comment on the matter, but a formal statement is expected later this afternoon.
Police in New Zealand contacted their family and confirmed their bodies had been found.
Hosking's physique had no longer been recovered, but she's amongst those presumed unimaginative, the newspaper reported.
Lisa Dallow with her husband Gavin on their wedding day in 2017. Ms Dallow remains in hospital with serious burns.
But a mid-morning update from geological agency GeoNet showed conditions on the volcanic island remained risky, which would likely delay any recovery. Twenty-eight people remain hospitalised, including 25 in critical condition.
"We'll miss him at the cricket -- I will, at least, and we'll miss him at the football".
"Our hearts break at the loss of Zoe at such a young age", he continued. "We know her loss will also devastate her school community and the local Girl Guides of which she was an active member", her step grandfather said.
At least six people have died since being rescued, police said.
"To correct an earlier statement, it is too early to confirm whether there will also be a criminal investigation".
Queensland Premier Annastacia Palaszczuk tweeted her condolences to the family, whom she said she knows personally.
The eruption of the volcano on White Island off the coast of Whakatane, New Zealand.
A 20-year-old Brisbane woman and her 47-year-old Mum have been formally identified as the first two Australian victims of Monday's volcano eruption disaster in New Zealand.
New Zealand's Prime Minister Jacinda Ardern speaks during an interview with Reuters in Wellington, New Zealand, December 11, 2019.
"Now we have the festive season, which will be celebrated by so many Queenslanders, but for this family, it will be one of deep poignancy".
Relatives of the pair are "united in grief", he said.
Nico Fournier of GNS Science said the chance of an eruption on Whakaari / White Island over the next 24 hours remained between 40 and 60 percent. "Eruptions in the next 24 hours are still likely to occur".
A plume of smoke could still be seen coming from the island.
"There are a number of people in hospital who can not communicate, they have significant burns not only to skin but internal organs", police minister Stuart Nash told Radio New Zealand. Eight others on the island are presumed dead.
GeoNet raised the alert level for the volcano in November because of an increase in volcanic activity.
GeoNet said in an early evening update that shallow magma within the volcano appeared to be driving the increased activity.