My interest in FDR caused me to become interested in Churchill. Most sources comment of the friendship between the two men, and their having the same goals in WWII.
In How Churchill Saved Civilization (an auspicious title indeed), John Harte mentions that FDR seemed at times to prefer Stalin, and that Churchill wanted to keep the British Empire intact, while FDR wanted countries of the Empire to become democracies.
They corresponded with each other even before the war, while Churchill was out of office. It seems that they got along very well on a personal level.
The problem that arose towards the end of the war was that Roosevelt didn’t seem to see the danger of Russia and Communism, and was inclined to trust Stain and take him at his word.
Churchill had always distrusted Stalin. A significant example was once when they were all chatting together informally after dinner, and Stalin said something to the effect that the best way to remove future threats from Germany at the end of the war would be to collect together the top-ranking 50,000 German officers and just shoot them all.
Churchill was highly offended, and stood up and walked out of the room. Roosevelt just laughed and didn’t take it seriously. Stalin then walked out after Churchill, apologised and said it was ‘just a joke’ and persuaded him to come back. Roosevelt thought he was overreacting. But of course we know that Stalin was fully capable of doing what he said, and was probably serious.
After the Warsaw Uprising in 1944, Churchill saw that Stalin could not be trusted in anything he said, and that Russia was determined to take over Poland and eastern Europe. He acted accordingly, but he could never convince Roosevelt (or the American generals, or American government in general) to take the threat seriously and try do something about it. They apparently saw Russia as a counter-balance to British power.
FDR had (in my opinion) an irrational hatred of “empire.” And since the British empire was an established fact–contrasted with the Soviet empire which at that point was merely a longed-for dream–FDR allowed that to influence his behavior.
Stalin killed off the Polish military elite in the Katyn Forrest. He killed off his own elites in the thirties. He was testing the waters with Churchill.