An investigation by BBC Radio 4 programme File on Four will disclose that the Department of Trade and Industry allowed a quantity of the metal, Beryllium, to be sold to Iran last year.
That metal is needed to make nuclear bombs.
Britain has had an arms embargo to Iran since 1993 and has signed up to an international protocol which bans the sale of Beryllium to named countries, including Iran.
MP's concerns
Liberal Democrat foreign affairs spokesman, Menzies Campbell, who has been alerted to the BBC programme's material, is said to be extremely alarmed.
Beryllium is a metal with a limited number of high-tech uses in civilian industry, but is mostly used in defence applications and is a vital component in a nuclear bomb.
The programme has also interviewed a leading nuclear weapons expert in the UK who says that the Beryllium and other items which the DTI has licensed to Iran add up to a shopping list for a nuclear weapons programme.
The UK has an arms embargo against Iran, but not a trade embargo.
Export control weaknesses
The programme highlights the weaknesses in the UK's new export control system, which was set up to stop the proliferation of nuclear weapons.
It will reveal that Iranian procurement agents have been working in the UK to get sensitive material back to Iran, and that Pakistan has also been successful in procuring material for its nuclear programme from here.
It is also likely to cause concern among Britain's allies.
President Bush named Iran as part of an "axis of evil" accusing the Iranian regime of sponsoring terrorism.
File on 4 is at 2000 BST on Tuesday 24 September on BBC Radio 4.
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