Powerline blog shared this back story that was new to me:
Mel Torme and Bob Wells wrote “The Christmas Song” in 1945 on a sultry day in southern California. Wells had written the first verse. Torme found it on the piano after he let himself in to Wells’s San Fernando Valley home for a songwriting session. When Wells turned up in tennis shorts and shirt (still looking hot, as Torme tells it), Torme asked him about “the little poem.” Wells told him, “It was so damn hot today, I thought I’d write something to cool myself off. All I could think of was Christmas and cold weather.” Forty-five minutes later they had produced the classic Christmas song. They promptly offered it to Nat “King” Cole; Cole fell in love with it on first listen. Because of his busy schedule, however, Cole didn’t get around to recording it until 1946.
That’s the story as we have all heard it. Telling this story in his memoir It Wasn’t All Velvet, however, Torme adds “a humorous footnote.” Cole had recorded the last line of the bridge as “To see if reindeers really know how to fly.” After the first pressings of Cole’s recording had turned the song into a hit, Torme and Wells pointed out Cole’s error to him: “Nat, a true gentleman and a dogged perfectionist, stewed over this mistake, and at the end of another recording session of his, with the same-size orchestra at hand, he rerecorded our song, properly singing ‘reindeer'” (in the version we all know).