The Government is considering whether people should be automatically put on the organ donor register unless they "opt out".
A system of presumed consent, which operates in several other countries including Sweden and Austria, has proved controversial in the UK.
The Chief Medical Officer for England, Sir Liam Donaldson, backs changing the law to drive up donation rates but some critics have argued against it, but his Scottish counterpart, Harry Burns, has said the public is not ready for such a system.
Health Secretary Alan Johnson will announce today that the Organ Donation Taskforce, set up in 2006 to look at barriers to organ donation, would examine the issue in detail.
The taskforce, lead by Elisabeth Buggins - chair of the NHS West Midlands strategic health authority - will focus on the moral and medical issues around presumed consent, including whether the family of somebody who has died should be given the final say on organs for donation.
She said: "I am very pleased that the taskforce has been asked to explore this incredibly important issue.
"We will establish a special sub group to take this work forward which will examine the complex medical, ethical, legal and societal issues."
Under the Human Tissue Act 2004, close relatives have the final say unless a person has actively put themselves on the organ donor register or expressed their wishes. But the family does not have the legal right to veto or overrule those wishes, even if they disagree.
More than 14.5 million people, about one in four of the population, are currently signed up to the organ donor register.
Mr Johnson said: "We know that around 8,000 people in the UK need an organ transplant but only 3,000 transplants are carried out each year.
"With more than 400 people dying every year waiting for a new kidney, heart, lung or liver, we need to do everything possible to increase organ donation.
"I want to see organ donation and transplant rates start to rise and match the rates seen in some other European countries, enabling us to save many more lives.
"This is a sensitive issue, but it is vital that all possible options for increasing the number of organs available for transplant are explored."
