Everyone should be seen as a potential organ donor on their death unless they expressly request not to be, England's chief medical officer says.
There is a major shortage of organs for transplant
Sir Liam Donaldson wants a system of "presumed consent" to be introduced in England to tackle organ shortages. His Scottish counterpart rejected the move.
One person a day dies after failing to find a suitable donor, data suggests.
The Tories opposed the move, saying it would be better to increase the number of people on the donation register.
Shadow health secretary Andrew Lansley said: "The state does not own our bodies or have a right to take organs after death."
He said the Conservatives were committed to enhancing organ retrieval teams in hospitals and appointing more donor liaison nurses as the best way to combat any "transplant crisis".
Sir Liam's Scottish counterpart, Harry Burns, rejected the idea for Scotland, saying there was no evidence that the public would support such a move.
A system of presumed consent was rejected by MPs when they voted on the Human Tissue Act in 2004.
Both the then Health Secretary John Reid and Health Minister Rosie Winterton declared it was not up to Parliament to make decisions about what became of people's bodies when they died.
But Liberal Democrat MP Dr Evan Harris - who chairs Parliament's all-party kidney group and introduced the Human Tissues Bill amendment, said on Tuesday that it was time for the government to back a change to presumed consent.
Sir Liam said that efforts to persuade more people to either carry donor cards or sign up to the NHS Organ Donor Register had failed.
Only 20% of the population - or 13 million - are on the register, despite the fact that surveys showed that as many as 70% of people wanted to donate their organs after death.
Because so many people die in a manner which makes them unsuitable donors, the pool of potential donors must be substantially larger than the numbers waiting for organs - currently nearly 8,400.
Sir Liam said any system would need proper safeguards
Sir Liam added that the gap between the number of people coming onto the waiting list and those coming off after receiving successful transplants was widening all the time.
Both these factors, he hoped, meant "MPs may change their minds".
The move has been welcomed by transplant campaigners and been endorsed by the British Medical Association (BMA).
"We must increase the number of donors available and the BMA believes that a system of presumed consent with safeguards, will help to achieve this," said Dr Tony Calland, chairman of the medical ethics committee.
Sir Liam said he was making the recommendation based in part on the experience of Spain.
Donation rates have almost doubled there to 35 people per 1m since a system of presumed consent was introduced in 1990.
Opt-out countries include:
Spain runs a so-called "soft" opt-out system, where even if the person has not themselves opted out of donation while alive, the views of relatives are sought and they can refuse consent.
Other countries, such as Austria, run a very strict system where the views of relatives are not taken into account at all.
After Vienna passed the presumed consent law in 1982, the donation rate quadrupled and by 1990 the number of kidney transplants performed was nearly equal to those on the waiting list.
However, Sweden - which also has an opt-out law - has a lower donation rate than the UK, suggesting that other issues may be at play.
It has been suggested that one of the reasons Spain's donation rate is so good is a result of the higher numbers of healthy people killed in road accidents.
Therefore even if the UK did choose an opt-out system, critics argue, its donation rates still might not rival those of Spain.
Sir Liam said that any system introduced in the UK would need "proper safeguards" and would likely take into account the wishes of close family.
He added that any new opt-out scheme would have to follow an intense public information campaign, and in any event, remained in the hands of parliament.
The Human Tissue Authority said the new act had only been in place for less than a year, and that it was not the right time to make changes to the legislation.
It believes a system of presumed consent for organ transplantation might undermine current provisions in the act for fully informed consent for other purposes, such as body donation for medical science or removal of tissue at autopsies.
Adrian McNeil, chief executive, said the option of presumed consent was debated extensively during the passage of the act through Parliament.
"It was decided that the opt-in system should remain. As long as the act is in force, the HTA will continue to support the requirement to obtain fully informed consent."