It was initially Franklin Roosevelt who allowed and made a decision to develop the atomic bombs to which President Truman finally decided to drop. The Manhattan Project was also part of the many initiatives by Franklin Roosevelt.
The only reason why Truman was left to the decision to drop the atomic bomb was because Franklin D. Roosevelt passed away while in office (after being elected to his fourth term). Before this, Roosevelt was the one who developed the program for the Manhattan Project and he was also responsible for navigating the US through most of World War II. Truman was only president for roughly 4 months before having to make the deicision about the atomic bomb.
