The UK now has a stockpile of 100 tonnes of plutonium - enough to make thousands of nuclear bombs, according to its national academy of science.
The plutonium could be used to power nuclear reactors
The Royal Society says the plutonium mainly comes from reprocessed spent uranium fuel from nuclear power plants.
The society warns that it could be made into "a crude nuclear bomb" by a terror group, and is calling for a strategy for its long-term use or disposal.
The government said stocks were protected against the risk of attack.
The society says it first highlighted the problem nine years ago.
And it warns that nothing has been done and that the size of the UK plutonium stockpile has doubled in that time.
Report working group chairman Professor Geoffrey Boulton said: "The stockpile has grown whilst international nuclear proliferation and terrorist threats have increased.
"Just over 6kg of plutonium was used in the bomb which devastated Nagasaki and the UK has many thousands of times that amount.
"We must take measures to ensure that this very dangerous material does not fall into the wrong hands."
Professor Boulton said the best option would be to convert the plutonium powder into pellets "which won't disperse in the atmosphere so readily".
"And the second thing you can do, and this might seem rather strange - you make it more radioactive," he added.
"More radioactive so that it's simply more difficult to handle and the ideal way to do that is of course to burn it."
The pellets could be used to power nuclear reactors, particularly if the government decided to build a new generation of nuclear power stations, he said.
The report added that, in the long term, the best way to dispose of the stockpile was to bury it deep underground once it had been burned as fuel.
A spokeswoman for the Department for Business, Enterprise and Regulatory Reform (BERR) said it welcomed the report.
The department agreed that a long-term strategy for dealing with the stockpile "should be considered as an integral part of energy and radioactive waste polices".
She said the Nuclear Decommissioning Authority (NDA) had advised the government on a range of options for managing the stockpile.
"It would not be appropriate to take any decisions until the public consultation on new nuclear [power stations] has been completed and the outcome is known," the spokeswoman said.
Earlier this month, environmental groups pulled out of the consultation on whether the UK should build nuclear power stations, saying it was not a "fair and full debate".
They had already won a High Court ruling that a previous consultation was "seriously flawed" and "misleading".
A public consultation is being held on the principle of including nuclear power as part of Britain's energy mix.
Most existing nuclear power stations are due to close by 2023 and the government has said its "preliminary view" is that new stations should be built - to reduce carbon emissions and Britain's reliance on foreign oil and gas imports.