Mr Browne said there were some 8,000 people waiting for organs.
A bill put forward by a Taunton MP, which aims to increase the number of organ donors in the UK, is being debated in the House of Commons.
Liberal Democrat Jeremy Browne wants to develop an opt out scheme where organs are automatically donated unless people have specifically requested not to.
He was inspired to act after meeting constituent Sean O'Brien who has liver cancer and is waiting for an organ.
About 1,000 people died every year waiting for a transplant, he added.
Introducing his Organ Donation (Presumed Consent) Bill from the backbenches, Mr Browne said: "We now have this position where a quarter of the population have signed the register and yet 90% say that, in principle, they have no objection to their organs being used."
He told the House there were currently some 8,000 people in the UK waiting for transplants and that figure was rising.
"It is a very pressing issue, it is a very real issue indeed for those people and their friends and families."
At present people must sign up to the organ donor register or their families must agree, before their organs can be used.
But Mr Browne said there would be no "cumbersome barriers" to people who wanted to opt out from the register.
"There would be an opportunity for you to opt out of giving some organs and not others or they could opt out altogether," he added.
He said his Bill had received cross-party support as well as from Chief Medical Officer Liam Donaldson and the British Medical Association.
Presumed consent would only apply to adults aged 18 or over.