The decision in 1958 to sell 20 tonnes of heavy water - a vital ingredient for the production of plutonium - appears to have been made by British civil servants with no input from ministers, the BBC report said.
The documents unearthed by BBC television's Newsnight programme, broadcast late on Wednesday, show British officials decided that it would be "over-zealous" to insist Israel use the heavy water only for peaceful purposes.
Previously, the US had refused to supply heavy water to Israel without such safeguards.
Heavy water is chemically the same as normal water, but where the hydrogen atoms are of the heavy isotope deuterium, in which the nucleus contains a neutron in addition to the proton found normally.
It can be used to turn natural uranium into plutonium, needed for nuclear weapons.
Israel's nuclear project
Archives in London reveal that the heavy water was shipped from Britain in Israeli ships in 1959 and 1960.
It was used for the production of plutonium at Israel's top secret Dimona nuclear reactor in the Negev desert.
The archives appear to show that the decision to sell the heavy water was taken only by civil servants, mainly in the Foreign Office and the UK Atomic Energy Authority, perhaps for economic reasons.
"The fact that Israel was trying to develop a nuclear bomb should not have come as any surprise... But that Britain should have supplied it with heavy water was indeed a surprise to me"
former US defence secretary
Newsnight said it had found no evidence that ministers in the then-British government of prime minister Harold Macmillan were ever consulted about the sale, or even told about it.
The records also show an explicit decision not to inform the United States, the report said.
Robert McNamara, who served as president John F Kennedy's defence secretary from 1961, shortly after the sale, told Newsnight he was "astonished" at the revelation.
"The fact that Israel was trying to develop a nuclear bomb should not have come as any surprise... But that Britain should have supplied it with heavy water was indeed a surprise to me," he said.
"It's very surprising to me that we weren't told because we shared information about the nuclear bomb very closely with the British."
After a British newspaper exposed the Israelis' work at Dimona in 1960, Britain refused a second Israeli request for more heavy water.