In September 1962, U.S. President Kennedy delivered his famous "We choose to go to the moon" speech, at Rice Stadium, promising to put a man on the moon before the end of the sixties. All the more famous, perhaps, because it ended up actually happening (with only five months to spare).
Less remembered, perhaps because it didn't end up happening, was Kennedy's 1963 speech to the United Nations. With the Cuban missile crisis less than a year behind him—and the threat of open nuclear war between Washington and Moscow still a fairly open question—Kennedy made an interesting suggestion: "Maybe the United States and the Soviet Union should team up and go to the moon together."
It was part of a broader speech about how maybe things had gotten a little out of hand, how perhaps threat of total nuclear annihilation all the time wasn't the best possible way to live. And, privately, it was at least partially motivated by how expensive getting to the moon was turning out to be: his initial request of $7-9 billion over the next five years would ultimately turn into $24 billion over seven.
Like Kennedy's other tentative olive branches, Khrushchev soon turned this one down. But according to Khrushchev's son, within a few weeks Premier Khrushchev was having second thoughts, and was privately warming to the idea of a joint US/USSR lunar mission. His son also said that, should space cooperation have panned out, Khrushchev was also making plans to step down the Soviet military, and shift from weapons production to commercial goods. Had Kennedy followed through on his proposed trip to Moscow in '64, the two leaders might well have "sealed the deal," and together ushered in a new era of international scientific cooperation and military deescalation.
But then Kennedy got shot in the head, and the Soviets didn't like or trust Johnson at all, and everybody basically hit the reset button on the whole "give cooperation a chance" thing.