The US president ordered troops into the Caribbean island, part of the Commonwealth, in October 1983 after prime minister Maurice Bishop was overthrown and murdered.
The move infuriated the British Prime Minister, who sent a response just after midnight on the day troops moved in, saying “This action will be seen as intervention by a Western country in the internal affairs of a small independent nation, however unattractive its regime.”
In tapes released by the White House of a phone call responding to this message, Reagan is heard speaking in a soft and apologetic tone as he seeks to appease his famously close ally.
Opening the conversation, he told Mrs Thatcher: “If I were there, Margaret, I’d throw my hat in the door before I came in.”
She responds, saying: “There’s no need to do that.”
In a conversation lasting more than ten minutes, the president explained at length that the United States had not given advance notice of the invasion because of fears a mole would leak sensitive information.
He added: “Incidentally, let me tell you that we were being so careful here that we didn’t even give a firm answer to the Caribbean states.
“We told them that we were planning, but we were so afraid of this source and what it would do; it could almost abort a mission, with the lives that could have endangered.
“When word came of your concerns - by the time I got it - the zero hour had passed, and our forces were on their way.”
He added that he had no “lack of confidence” in the UK’s ability to keep a secret, adding: “It’s at our end.”
Speaking in a calm voice, Thatcher seems to accept the apology and draws on her own experience from a year earlier in empathising with Reagan.
She says: “I know about sensitivity, because of the Falklands.
“That’s why I would not speak for very long even on the secret telephone to you.
“Because even that can be broken. I’m very much aware of sensitivities. The action is under way now and we just hope it will be successful.”
She ended the call by saying she had to return to a “tricky” House of Commons debate.
Reagan, whose mood seems to have lifted at this point, answers: “All right. Go get ‘em. Eat ‘em alive.”
The recordings from the White House Situation Room were released to author William Doyle following a Freedom of Information Act request he made in 1996.
He released extracts through the New York Post.
Doyle, author of Navy Seals: Their Untold Story, said Reagan recorded many of his phone calls with heads of state so that an accurate record could be kept.
He told the newspaper: “Until now, taping was thought to have stopped in the Nixon era. I discovered that was not the case.
“There is no evidence that I have seen that Reagan audiotaped any other closed-door non-public White House business.”
Talking about the Thatcher call, he added: “It is a remarkable, secret presidential call, as Reagan is in the middle of a major presidential crisis, thousands of US troops are in motion into combat and he is talking without a script.”
Other recordings include the president attempting to persuade Israeli prime minister Menachem Begin to delay pulling troops from Lebanon in 1983; discussing the release of western hostages in the Middle East with Pakistani president Muhammad Zia-ul-Haq; and a conversation with Syrian president Hafez al-Assad, the father of Syria’s current president Bashar al-Assad.
Grenada gained independence from the UK in 1974.
Operation Urgent Fury was launched by the US after a power struggle within Bishop’s leftist New Jewel Movement led to his arrest and killing.
The invasion resulted in a US victory within weeks.