As British troops closed in on final victory, Mr Reagan made a late-night phone call to Mrs Thatcher urging her not to completely humiliate the Argentines.
However, his request fell on deaf ears as the Prime Minister insisted she had not sent a British task force across the globe just "to hand over the Queen's islands to a contact group".
Mr Reagan made his call to Mrs Thatcher in Downing Street at 11.30pm London time on May 31 as British forces were beginning the battle for Port Stanley, the Falklands capital.
The Americans had already proposed sending a US-Brazilian peacekeeping mission, and the president suggested the time had come to show magnanimity.
According to the official No 10 note, Mrs Thatcher told him: "Britain had not lost precious lives in battle and sent an enormous task force to hand over the Queen's islands to a contact group.
"The prime minister asked the president to put himself in her position.
"She had lost valuable British ships and invaluable British lives. She was sure the president would act in the same way if Alaska had been similarly threatened."
Mrs Thatcher said "the most sensible thing" would be for the Argentinians to withdraw, before ending the conversation with a familiar refrain: "There was no alternative."
As the battle reached its climax she even drafted a telegram to the Argentinian leader General Galtieri – although it was never sent – demanding for a final time he withdraw his forces.
"In a few days the British flag will be flying over Port Stanley. In a few days also your eyes and mine will be reading the casualty lists," she wrote.
"On my side, grief will be tempered by the knowledge that these men died for freedom, justice and the rule of law. And on your side? Only you can answer that question."
It was not the only time during the conflict the UK had problems with her closest ally.
On April 21, as the British task force approached the islands, US Secretary Of State Al Haig – who had been trying frantically to broker a settlement – spoke to the British ambassador to Washington, Sir Nicholas Henderson.
He told him he intended to inform the Argentinian junta that UK troops would be landing on South Georgia, the first of the islands to be seized by the Argentinians.
Mr Henderson was appalled. He told Mr Haig he was going far beyond the obligations of a neutral negotiator and the information could be used by the Argentinians to mount a submarine or suicide air attack on the task force. Reluctantly, Mr Haig promised to keep quiet.
The war ended on June 14 when UK troops entered Port Stanley.