You have two very different questions here — one in the title and another in the comment.

The title question is easier to answer. In the summer of 1945 the Japanese did try to send out "peace feelers" to the Soviet Union. The USSR was neutral with regards to the Japanese, and there were elements of the high command that thought they might be able to serve as mediators to negotiate a diplomatic end of the war with the other Allies. In Tokyo the Japanese made approaches on the Soviet ambassador to Japan, and in Moscow the Japanese ambassador attempted to get meetings with Molotov to make the suggestion. Neither went anywhere — the Japanese were both a bit too timid in their approaches (it was supported by the Emperor but not really an official overture, they were still feeling it out), and the Soviets rebuffed them effectively (because Stalin had agreed at Yalta to declare war on the Japanese and was looking forward to acquiring key territorial concessions as a result). As the Japanese terms were never formally expressed, we cannot really know what they would have been, but it was clear that the major terms involved the disposition of the Emperor and Empire after the war, and possibly some territorial disputes. The US was aware of these negotiations because it had cracked Japan's diplomatic codes, and used the insights gained to consider their options at the end of the war. It should be made clear that only a few elements of the Japanese high command were pursuing this "peace" strategy; there was also a "military" wing that was more committed to a final, bloody showdown. On this topic see esp. Hasegawa's Racing the Enemy. The most important historical point here is that the US knew that the Japanese were hung up on the implications for the Emperor as a result of "unconditional surrender," and that Truman and Byrnes deliberately rejected suggestions (from such non-peaceniks as the Secretary of War and Winston Churchill) that they clarify this in an attempt to make it easier for the Japanese to surrender. (This is discussed in my link below.)

As for your second question, the notion that there were only two options — "use two atomic bombs on cities within three days of one another" and "have a ruinous invasion that will kill millions" is indeed a false choice. There were more alternatives on the table at the time. I have written a bit on these here — these are all alternatives that were considered by people at the time, not just retrospective analyses. To point alternatives is not to imply that the alternatives would have resulted in a better world, or were morally required to have been explored more deeply. But it does do a good job of highlighting the choices and contingency of the whole thing. The "bomb or invade" narrative is a postwar one meant to justify the bombing; it is not at all how they thought of it at the time (they saw it as "bomb and invade").