Both.

The United States' relations with the USSR were never good - this was a country which had intervened in the Russian Civil War on the losing side while the head of its socialist party was in prison on sedition charges, and the First Red Scare was contemporaneous with the Russian Civil War. The United States didn't even recognize the Soviet Union until 1933, and while no communists and few socialists won elections in this period, FDR's detractors were all too glad to call him a communist as a criticism.

But the immediate roots of heightened confrontation between the US and USSR are certainly in the aftermath of WWII, because this conflict significantly increased the global influence of both nations, and the first areas of tension in the Cold War included the Greek Civil War (a war fought between an army associated primarily with the WWII-era government in exile, and communists associated primarily with WWII partisan movements) and the blockade of the non-Soviet parts of Berlin, which is Germany's capital and was divided into sectors by the victorious allies.