Files released by the National Archives at Kew, south-west London, under the 30-year rule show that as British troops closed in on final victory, the US president 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 a defiant prime minister insisted that 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 Downing Street at 11.30pm London time on May 31, 1982, as British forces were beginning the battle for Port Stanley, the Falklands capital.
The Americans had already proposed sending a joint US-Brazilian peacekeeping mission, and the president suggested the time had come to show magnanimity.
"The best chance for peace was before complete Argentine humiliation," he told her.
"As the UK now had the upper hand militarily, it should strike a deal now."
Mrs Thatcher was having none of it. The United Kingdom, she said, could not contemplate a ceasefire without Argentinian withdrawal.
According to the official No 10 note, she 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.
"As Britain had had to go into the islands alone, with no outside help, she could not now let the invader gain from his aggression. 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 that the president would act in the same way if Alaska had been similarly threatened."
The prime minister said "the most sensible thing" would be for the Argentinians to withdraw, before she ended 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 that 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 that Britain had problems with her closest ally.
On April 21, as the British task force approached the islands, US secretary of state Al Haig told the British ambassador to Washington, Sir Nicholas Henderson, 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 Argentines.
"If the Americans acted in this way, they would be able to show even-handedness to the Argentinians and this would enable them to continue their role as go-between," Mr Haig argued.
The ambassador was appalled. He told Mr Haig he was going far beyond the obligations of a neutral negotiator and that the information could be used by the Argentines to mount a submarine or suicide air attack on the task force.
Reluctantly, Mr Haig promised to keep quiet.
Overall, however, Mr Henderson concluded that Britain had cause to be grateful to Mr Haig for ensuring a divided Reagan administration ultimately came down on the side of the UK.