There is, as you may know, a historical argument that one of the reasons for dropping the atomic bomb on Japan was to send a message to the USSR. While the evidence that this was the primary reason for using the bomb is fairly thin, some members of Truman's cabinet (notably Secretary of State Byrnes) clearly saw this as one of the several benefits to using the bombs against Japan.

The US did draw up plans as to what a nuclear attack against the USSR would look like, and how it would be launched, as early as late August/early September 1945. So this was not out of the range of the strategic imagination, despite the fact that the US and USSR were Allies against the Axis powers.

But the US had a very limited supply of nuclear weapons, and relatively limited means of delivering them, well up until the early 1950s. By the 1950s the USA could have significantly damaged or destroyed the USSR with nuclear weapons had they chosen to, but it would have come with heavy costs. The USSR, by around 1950, had access to nuclear weapons as well, but their means of delivering them were mostly limited to Europe — so they could have caused a lot of damage to US allies in the area, but it would have been very hard for them to threaten the US homeland. Still, that is enough of a threat that the US was not interested in preempting a nuclear war.

If the US had openly threatened to use nuclear weapons against the USSR in 1945, it would have likely been seen as the bluff it was (the USSR had some idea of how few weapons the USSR had, because of its spies), and would have definitely alienated the US with many of its European and Asian allies who were eager to stop fighting (and were more pro-Soviet than were most Americans). A nuclear war with the USSR in 1945 would have been more like World War II — lots of conventional action with a few atomic bombs — and less like what we think of as World War III (e.g. all atomic bombs). It would have been messy and nobody was interested in it — the US public was ready for a peacetime recovery.