In my high school history class we debated whether or not to drop nukes on the Japanese if we had the option presented to us and we did a similar thing in sophomore English when we read the book Hiroshima. Often times in the debate, people presented the "fact" that factions within the government were moving towards a peace agreement with the US/Allies. But nothing ever went into specific actions or people, just a vague statement of like that. Is there any proof or evidence of this?
Well, it's first worth emphasizing that they were not "moving towards a peace agreement with the US/Allies." It's more than there were factions of the Japanese high command who were exploring the possibility of getting the (then-neutral with regards to the Pacific War) Soviet Union to intercede with the other Allies (chiefly the US, UK, and China) to remove the requirement of "unconditional surrender" and replace it with something that was more palatable to the Japanese. Specifically the Japanese were concerned that "unconditional surrender" did not clarify whether the Emperor system (the kokutai) would persist into the postwar. There were other issues as well but this was the chief sticking point.
We have lots of evidence for this in the postwar (e.g. from Japanese archives and records), but the most interesting evidence for this is in the form of the MAGIC cables. These were intercepted and decrypted communications between the Japanese Foreign Minister in Tokyo and his Ambassador to the USSR in Moscow. The US had broken the Japanese diplomatic code and was able to read these more or less in real-time, and so knew much about the internal schisms within the Japanese high command and their attempts to maneuver with the Soviets. Even if they had not decrypted the intercepts, the Soviets told the Americans and British about these attempts at the Potsdam Conference.
So it's more nuanced than just saying the Japanese were looking for peace. They were attempting to set up a separate negotiation with the USSR, and doing it to increase their own postwar position. Now one can interpret this in different ways, and the Allies certainly did. Some, including the US Secretary of War and even Winston Churchill, saw this as a call for clarifying or lessening the demand for "unconditional surrender," on the theory that maybe if they made clear the Emperor would not be prosecuted, or allowed to remain a ceremonial figurehead, that this would cause an immediate surrender. Others, including the US Secretary of State, saw the cables are further evidence the Japanese high command still were not ready to admit their utter defeat and thus the atomic bombs and perhaps invasion were still necessary.
The best book on the Japanese diplomatic attempts to end the war is Tsuyoshi Hasegawa's Racing the Enemy: Truman, Stalin, and the Surrender of Japan. It is an interesting read, with more intrigue among the high commands of Japan, the USA, and the USSR than any Game of Thrones episode.