Gordon Brown is sympathetic to calls for hospitals to be allowed to remove organs from dead patients without explicit consent, but families would have a veto which would allow them to stop organs being used to save the lives of others if, as expected, the Government brings in a system of "presumed consent".
People could also opt out of the national register during their lives, in contrast to the current system of opting in and carrying a donor card.
Officials said Mr Brown had an "open mind" about the idea, but wanted a national debate before the Government takes a decision. A taskforce on organ donations will consider the proposal later this year.
Writing in The Sunday Telegraph, Mr Brown said the "presumed consent" system used in Spain "seems to have the potential to close the aching gap between the potential benefits of transplant surgery in the UK and the limits imposed by our current system of consent".
The Health minister Ben Bradshaw said the proposal could save thousands of lives and reduce medical costs. He said: "Given that 1,000 people in this country die every year waiting for organs, and we have a relatively low level of donation, anything we can do to get those numbers up must be right." He said that 90 per cent of the population were in favour of organ donation and only 20 per cent on the register, but acknowledged that it was a "delicate issue".
Joyce Robin, of the watchdog Patient Concern, said: "They call it presumed consent, but it is no consent at all," said "They are relying on inertia and ignorance to get the results they want."