The other bodies will be repatriated later, the outlet said.
On October 23, 39 bodies were discovered in the back of a lorry that had arrived into Grays in Essex outside of London.
The family alerted authorities that the victims could be Vietnamese nationals, as United Kingdom police initially believed all 39 victims to be Chinese.
However, the families will have to pay up to €2,569 for the repatriation, according to a statement seen by the Reuters news agency.
The website published photos of ambulances that transported the bodies to their home provinces south of Hanoi.
The British ambassador to Vietnam, Gareth Ward, said that the two countries will continue to work together "to prevent human trafficking and protect vulnerable people here".
The other remains are expected to arrive in the coming days, though officials have not announced the schedule.
Hoang Lanh, father of 18-year-old victim Hoáng Van Tiep, said he had been informed by a local government official that his son's remains would be brought home Wednesday, along with those of six other victims from Dien Chau district in Nghe An province, including Tiep's cousin Nguyen Van Hung.
Victims' families have taken out loans from the Vietnamese government to repatriate their loved ones' bodies, €1,635 for ashes, or €2597 for a coffin carrying the body.
"We have been waiting for this moment for very long time".
"We are deeply saddened, but we have to hold back the emotion to organize the funeral for my son", Gia said by phone from his home in Can Loc, Ha Tinh province.
Most hailed from just a handful of central Vietnamese provinces, which are among the poorest in the country and where well-entrenched networks of brokers can easily facilitate risky trips overseas.
Ten of the dead were teenagers, including two 15-year-old boys.
They have been taken to their families after a flight landed in Hanoi.
Many arrange trips through brokers who promise them well-paid jobs, and end up working in nail bars or on cannabis farms, heavily indebted to the smugglers who organise their trips.
Robinson, from Northern Ireland, pleaded guilty to conspiring to assist illegal immigration.
The repatriation came after a man accused of forcing the truck pleaded guilty to helping in plotting illegal authorities in British courtroom.
Eight people have been arrested in connection with the case in Vietnam, while the driver of the truck, Maurice "Mo" Robinson, 25, has been charged with 39 counts of manslaughter in the UK.
