Jeremy Corbyn has urged MPs to back a change in the law on organ donation that would regard people as giving their consent unless they actively opt out.
However, the so-called "deemed consent" bill, brought forward by Labour MP Geoffrey Robinson, would reverse the existing system so that all patients are assumed to be willing organ donors unless they have stated otherwise.
Mr Robinson said a cautious assessment by the NHS suggested that the opt-out system, backed up with the right resources, could save up to 500 lives every year.
In July the government said it would consult on introducing such a system.
Although the number of registered organ donors has risen from 14.1 million people to 23.6 million in the past decade, around 6,300 Britons are now on the waiting list for a transplant.
"We have some of the lowest rates of consent for organ donation in western Europe, low family rates of consent being one of the major barriers".More news: 4 players to watch at the Honda Classic
They also heard a grieving father's decision to allow his nine-year-old daughter's organs to be used to save four lives should serve as "inspiration" to others.
Labour and Conservative MPs put aside party differences to sort out the Bill.
"Who are we in this House to say that if we can prolong life and improve the quality of life of people that suffer from rare diseases, for example, such as John, then I think we should do so", she added.
She said: "We are supporting this Bill, we are determined to ensure that we secure more organs available for transplant, because we are very concerned that we are losing lives unnecessarily".
Under current legislation, those willing to donate their organs after death must indicate their consent by signing a national registry through their GP surgery, hospital or on the NHS organ donation website.