Truman definitely thought that the Soviet declaration of war on Japan might lead to their surrender. This is reflected in the journals he kept at the time. He also knew that the Japanese were trying to court the Soviets to act as a neutral negotiator with the United States — because the US had broken the Japanese diplomatic codes and were reading their mail (the MAGIC decrypts).

What this framing misses, though, is how Truman and his cabinet understood these things. Byrnes, his Secretary of State and perhaps the most influential person on him regarding foreign policy at Potsdam, thought that the MAGIC decrypts showed that the Japanese were, after all of their defeats, not ready to accept the unconditional surrender the Americans wanted. So he saw them as reinforcing the need for the atomic bomb.

Separately, the atomic bomb gave Truman and his confidantes the confidence that they might not need the Soviets to have an end to the war. He deliberately set the timing of the first atomic bomb with an aim to pre-empting the Soviet invasion, which he believed was to take place mid-August, which — if the bombs did cause the Japanese to surrender, would lock Stalin out of several key pieces of territory he was getting in exchange for declaring war.

Overall, your question makes it sound like they were actually weighing whether to use the atomic bomb. They weren't. They knew they wanted to use them; they had multiple reasons for thinking that using them was a good idea for the short and long term. They weren't looking for a way to get Japan to surrender without using the bombs; they were if anything eager to use them.

Your question is ambiguous about the timing, but it should be made clear that the Soviet declaration of war came after Hiroshima. It was, in fact, moved up a bit by Stalin, because he too feared that the Pacific war might end before he got involved in it (and he wanted the aforementioned territory).

All of which is to say, there was a lot of politicking going around on the US and Soviet sides during this, and both knew the Japanese were in a position where the war could possibly be over fairly soon. Despite this, the Americans were actually quite surprised it ended as soon as it did. It was a chaotic and messy period, and nobody was clairvoyant.

Tsuyoshi Hasegawa's Racing the Enemy is the best book for the three-way story (US, Soviet, Japanese) of the end of the war; it's complex and fascinating.

There's a number of caveats to this.

The first is the Potsdam Declaration, issued by the US, Britain, and China on July 26 1945. The USSR was not a signatory at the time due to a nonaggression pact with Japan, but was appraised of the arrangements before hand. The Potsdam Declaration was in actuality a fairly ambiguous document. This was purposeful. It called for a number of general things, namely that Japan would surrender territory it seized, turn over war criminals, and total disarmament. It assured Japan that its home territories would remain congruous, that Japanese people would not be enslaved, and that Japan would not become a permanent colony of an overseas power.

The Allies understood the document as calling for an end of military rule, restoration of democratic government, and the unconditional surrender of the Empire of Japan. It's ambiguity was purposeful as the Allies still weren't sure at the time exactly what would happen if Japan actually surrendered.

Japan hotly debated these terms when they received them, and there was a willingness to accept them. The main issues were two fold;

1, the Emperor. The Potsdam Declaration was vague. The Japanese weren't sure if the Allies were expected to dethrone the Emperor or prosecute him as a leader of the country. This was a serious sticking point, especially for the culturally conservative members of the Imperial military.

2, no surrender. While historians have pushed back on the idea that Japan was all 'fight or die', these was a serious element of the military culture of Japan, especially the Army. Remember that even when the Emperor ultimately chose to surrender members of the Army tried to stop him during the Kyujo Incident.

Truman's role in all this is often overstated. He was fairly inexperienced in foreign policy and was largely merely continuing the course set by his predecessor. The Japanese debated but did not respond to the Potsdam Declaration. They didn't know the USSR was already appraised of it and wanted to try and see if a separate peace with the Soviets was possible. While many in the government wanted to accept, Japanese newspapers reported on July 28 that it wasn't being accepted. There's a bit of debate here as the Japanese phrase used can mean both 'no comment' and 'refused' (more or less).

The lack of official response simply meant the Allies continued the war.

On August 6, Hiroshima was struck with the first atomic bomb. The second hit Nagasaki on August 9. While the US had long broken Japanese diplomatic and naval codes, it's unclear how much they knew about the internal debate over surrender. I suspect American war planners simply didn't care. The war wasn't over. The bombs were dropped.

Contrary to criticism of Truman, he didn't make this decision unilaterally. The US had a secret pact with Britain that both powers would only use atomic weapons with permission from the other. The US requested as the pact ordained and permission was given. While there there is ongoing debate about the fine details of the dropping of the bomb, I think it's mostly overstated. The war was long and the US and Britain simply wanted it to be over. In their eyes they'd issued the terms of surrender and all that was left was the waging of war till it came.

The USSR declared war on Japan August 8. On August 9th the Big Six - Japan's high command - was evenly split on surrender with no consensus. This same day, the USSR invaded Manchuria. Hirohito broke this deadlock and on August 10 Japan issued the Allies a qualified acceptance of the Potsdam Declaration with the caveat that the Emperor would remain the Emperor. The Allies accepted this one term on August 12.