Yesterday was the 70 anniversary of the bombing of Hiroshima. It is often stated that the bombing was carried out in order to limit casualties that an invasion would have caused. Were there any attempts at a diplomatic solution before the bombing?

There was no serious attempt at a diplomatic solution before the bomb. In fact, some deliberate actions were taken to scuttle a possible diplomatic solution.

The US knew (because they were able to intercept and decrypt their communications) that the some members of the Japanese high command were interested in finding a diplomatic solution. They knew that the part the Japanese were finding hardest to swallow was the demand for "unconditional surrender," because it did not clarify what the role of the Emperor would be in the postwar, or whether the Emperor would be tried as a war criminal. The Japanese saw the Emperor system (the kokutai) as being essential to the definition of what it meant to be "Japan," similar to the way that Americans view the US Constitution as part of the definition of what "the United States of America" means. So this was a significant stumbling block.

The US was planning to issue a proclamation urging the Japanese to surrender and promising more ruin if they did not — the famous Potsdam Proclamation. There was intense debate within the US high circles about how this should be worded, and whether they ought to clarify their position regarding the Emperor and Japan's postwar government.

The US War Department's Operational Division thought the Potsdam Proclamation ought to include the line: "The Japanese people will be free to choose whether they shall retain their Emperor as a constitutional monarchy." They wanted to do this because, as they explained:

The primary intention in issuing the proclamation is to induce Japan’s surrender and thus avoid the heavy casualties implied in a fight to the finish. It is almost universally accepted that the basic point on which acceptance of surrender terms will hinge lies in the question of the disposition of the Emperor and his dynasty. Therefore, from the military point of view it seems necessary to state unequivocally what we intend to do with regard to the Emperor.

The US Joint Chiefs of Staff thought the following line might do the trick: "Subject to suitable guarantee against further acts of aggression, the Japanese people will be free to choose their own form of government."

Both of these suggestions got nixed. So did attempts by the Secretary of War, Henry Stimson, to add lines clarifying the position of the Emperor. So also was a similar suggestion by none other than Winston Churchill. I only point out these names because sometimes people like to claim that the idea of modifying the surrender requirements is some kind of left-wing, anti-war, pro-Japanese sort of sentiment that only an outsider might have believed was a good idea. These guys are are inside as it gets.

The biggest opponent to modifying the Potsdam Proclamation was James Byrnes, the newly-appointed Secretary of State. It seems likely that Byrnes not only expected the Japanese to reject a Potsdam Proclamation that didn't clarify the role of the Emperor, but that he wanted them to. He looked forward to using the atomic bombs, because there would be, in his mind, a lot of benefits from them. They might end the war on better terms, they might end it quickly, and they might influence postwar relations, especially with the Soviet Union. (Sometimes the fact that Byrnes thought this is taken by some to imply that the US only cared about the USSR with respect to the bombs. This is not the case — but it was seen as an additional benefit, by some, but not all. There were many "up sides" to using the bombs.)

Truman himself seemed to go along with Byrnes on this. He did not, at the time, articulate why explicitly, except once. When Churchill asked about maybe modifying the terms of surrender, to help the Japanese preserve their honor (which would make them more likely to accept it, given their cultural associations with honor), Truman replied that the Japanese had no honor since Pearl Harbor. Which, to my mind, makes it look like Truman was doing this in a somewhat vindictive, punitive way — he wanted the Japanese to grovel.

As it was, even the atomic bombs and the invasion of Manchuria by the USSR did not make the Japanese fully embrace unconditional surrender immediately — it took several days of diplomatic back-and-forth before they finally agreed, with the Emperor himself taking responsibility for it. The US did not, in the end, try the Emperor as a war criminal, and allowed him to maintain a symbolic role in the new Japanese government.

If you are interested in all of this diplomatic back and forth (and there is a lot, and it is extremely interesting) between the US, the Japanese, and the USSR, Tsuyoshi Hasegawa's Racing the Enemy is a great account of that, whatever one thinks of his ultimate thesis (that the bombs were both unnecessary and did not actually end the war).

I wrote an article related to this question earlier this week, and as you can see there is some debate/discussion of it in the comments section (some of my responses I lifted and modified in my response here, for the sake of conserved effort).