The UK has one of the lowest organ donation rates in Europe.
There are currently around 8,000 people in the UK who need an organ transplant but only 3,000 operations are carried out each year.
Every year, 1,000 people in the UK die after waiting for a transplant.
Currently, people must sign up to the organ donor register - or their families must agree - before their organs can be used.
Health Secretary Alan Johnson announced on Monday that £4.5m would be spent on a public education campaign aimed at signing up 20 million people to the organ donor register by 2010, and 25 million by 2013. This would double the numbers currently on the register.
A similar campaign has already has some success in Scotland.
However, Mr Brown said that if the campaign failed to have the necessary impact a switch to presumed consent was on the cards.
He said: "I'm not ruling out a further change in the law.
"We will revisit this when we find out how successful the next stage of the campaign has been."
Mr Brown's comments were echoed by Nicola Sturgeon, the health minister in Scotland, who said the situation would be re-assessed in 2013.
NHS Scotland has been asked to produce new awareness-raising material to inform people about the existing legislation.
Elisabeth Buggins, chair of the taskforce, said the issue raised "really strong emotions."
But she said the task force, which had gathered over 400 pages of evidence from across the world, had found little evidence that presumed consent would increase the number of organs available for transplant.
She said although a system of presumed consent in Spain had been followed by a rise in organ donation, that rise was not thought to be down to the switch.
Ms Buggins said: "We found from recipient families and donor families that the concept of gift was very important to them and presumed consent would undermine that concept.
"We also found that it has the potential to erode trust in doctors, and we know that is very important to the levels of donation."
Instead, Ms Buggins said a rise in organ donations was more likely to be achieved by increasing in the number of donor coordinators who work with bereaved families, and the number of specialists who retrieve organs, and by launching public information campaigns.
"There is lots of fear out there that organs are taken from patients before they are dead - that is absolutely not true."
The medical profession is divided on the issue of presumed consent.
In September, intensive care doctors told the BBC they were deeply concerned about any radical changes to the law on organ donation.
We know that the majority of the population want to be organ donors, but only 25% are on the register
Dr Vivienne Nathanson British Medical Association
Research by the Intensive Care Society suggests many specialists are worried that such a move would damage the trust between patients and doctors.
But Dr Vivienne Nathanson, chair of ethics at the British Medical Association, said she was disappointed by the task force's findings.
She said presumed consent was not a panacea, but was likely to result in a 10-15% increase in donated organs - if sufficient surgeons, intensive care beds and transplant coordinators were put in place.
Gordon Brown continues to support 'presumed consent'
She said it would also encourage families to discuss their views, and make their position clear.
The BMA supports a 'soft' system of presumed consent, where individuals who do not want to donate their organs have a formal mechanism for registering that objection and where families are consulted to identify any unregistered objection.
Dr Nathanson said: "We know that the majority of the population want to be organ donors, but only 25% are on the register.
"Turning it round the other way, so that you take organs from everyone unless they have either put their name on a register to opt out, or their family say they wanted to, but haven't got round to it, means you could quite significantly increase the numbers of donors."
Lack of capacity
But Tim Statham of the National Kidney Federation said organs were being wasted because of a lack of capacity in the NHS - a situation which presumed consent would not solve.
And Professor John Fabre, a former president of the British Transplantation Society, said presumed consent was a "simplistic" way to try to boost organ donation rates, which would be a "waste of time".
Joyce Robins, co-director of Patient Concern, said: 'We can only hope that Gordon Brown does not follow the example of the Welsh Health minister Edwina Hart, who rejected the all-party Welsh Assembly report when they decided against presumed consent after weeks of evidence and is still pressing ahead."
A report recommending a radical overhaul of the UK organ donor network in a bid to double the number of organs available for transplant, has already been published by the Organ Donor Taskforce, and is being implemented.
The latest recommendations on presumed consent are not binding and the government could decide to press ahead with changes to the legislation.
Any change to the system would involve amending the Human Tissue Act of 2004.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.