Some relatives told the BBC they have been approached by the Vietnamese authorities with consent forms.
These gave them the option of taking a loan from the government, or covering the repatriation costs themselves.
The BBC has contacted the Vietnamese embassy in the UK for comment.
Authorities found the bodies of 39 Vietnamese nationals inside a container in the town of Grays in Essex last month. Two lorry drivers have since been charged with manslaughter, and several other men have been arrested in connection with the case.
While the police identified all the victims ten days ago, the bodies have not yet been returned to Vietnam.
Coroners in England and Wales must grant permission before a body can be moved abroad or to other parts of the UK.
What has the Vietnamese government offered?
Consent forms seen by the BBC say: "I, on behalf of the family, pledge to pay back all relevant expenses that the authorities have made in advance in order to repatriate them from the UK."
Victims' families have been offered a discounted price of 66 million dong (£2,200; $2,840) to repatriate a body or 44 million dong to bring back the ashes, the BBC has learnt.
Families had already taken out loans of up to £40,000 to pay for smuggling their relatives to the UK.
Pham Ngoc Tuan, brother of one of the victims. told the BBC's Nga Pham: "We already had to borrow so much money, we had to mortgage what we had. I don't even know if we can borrow anymore."
"We have been told the main issue now is money and even though we had to borrow a lot of money to send them away, we may borrow more to bring them home," said Tien Pham, cousin of two victims, in a Facebook live stream.
In the wake of the deaths, the Vietnamese embassy in London pledged to "closely co-ordinate with the relevant authorities of Vietnam and UK to support the families of the Vietnamese victims... to bring their loved ones home".
It also issued its "heartfelt condolences" to the families.