The date was May 1963 and the year before China had attacked India along its 2,000-mile Himalayan border, overpowering and defeating the poorly trained and badly equipped Indian troops. At dispute were two areas under Indian control, Aksai Chin in Ladakh and another area on the north-east frontier.
When Mr Kennedy and his senior officials met in the White House, a ceasefire had been called between China and India, with each side having lost 500 troops. But the US president and his advisers discussed the possibility that China might attack again and how they should respond to requests for help from Indian Prime Minster Jawaharlal Nehru.
On the tape, made public this week, Robert McNamara, who was then Mr Kennedy's defence secretary, says: "Before any substantial commitment to defend India against China is given, we should recognise that to carry out that commitment against any substantial Chinese attack, we would have to use nuclear weapons. Any large Chinese Communist attack on any part of that area would require the use of nuclear weapons by the US and this is to be preferred over the introduction of large numbers of US soldiers."
Moments later, having listened to Mr McNamara and others, Mr Kennedy says: "We should defend India, and therefore we will defend India." He does not specify whether he would authorise a nuclear strike and some analysts have said such an option would have been dismissed the next year when China tested its first nuclear weapon.
Maura Porter, archivist for the John F Kennedy Presidential Library and Museum in Boston, which released the recordings, said they offered researchers and historians "a unique perspective as to the inner workings of the Kennedy White House". She added: "When one listens to this recording and others at the Kennedy Library, they hear first-hand how critical national security matters were debated and discussed." Stephen Cohen, a senior fellow at the Brookings Institution in Washington, said the US had long been warning India to be wary of China's expansionist intentions. "[In 1962] they came in and swept the board clean in a brilliantly planned invasion. Nehru was in a panic. He gave a speech in which he basically said to those Indians [in areas then occupied by Chinese forces] good luck but goodbye; we cannot defend you. He was desperate and wrote letters to Washington asking for our help."
Mr Cohen believes that Mr Kennedy's senior officials may have raised the nuclear option to deter the president from getting involved on India's behalf.
On the tape, General Maxwell Taylor, then chairman of the joint chiefs of staff, tells Mr Kennedy: "This is just one spectacular aspect of the overall problem of how to cope with Red China politically and militarily in the next decade. I would hate to think we would fight this on the ground in a non-nuclear way." Mr McNamara told the International Herald Tribune that he could not remember the conversation but that the recording "is probably correct".