Truman's own diary would be a pretty good source: https://nsarchive.gwu.edu/document/28466-document-47-trumans...
The diary is full of revealing quotes like "Discussed Manhattan (it is a success). Decided to tell Stalin about it. Stalin had told P.M. of telegram from Jap Emperor [actually from the Foreign Secretary] asking for peace. Stalin also read his answer to me. It was satisfactory. Believe Japs will fold up before Russia comes in. I am sure they will when Manhattan [A-bomb] appears over their homeland. I shall inform Stalin about it at an opportune time."
For some timeline the US had secretly arranged for Russia to declare war on Japan on August 8th. We nuked them first on August 6th, and then again on the 9th. The goal was to get them to surrender to the US before the Soviets made significant inroads in Japan.