(1) February 1945, when Japan's tentative negotiators determined that "the Soviet Union would not hesitate to intervene militarily in the Far East once the situation turned favorable in Europe" (despite its earlier Neutrality Treaty with Japan)
(2) Early June 1945, "when the showdown Battle of Okinawa had been lost, when government analyses indicated that the war effort could soon continue no longer, and when General Umezu unveiled for the emperor the bleak results of his personal survey of the situation in China"
(3) Late July 1945:
Their third missed opportunity was July 27-28, when the Potsdam Declaration arrived and the Suzuki cabinet, after careful deliberation, twice publicly rejected it. At the time no member of the "peace faction" came forward with a proposal for accepting the Potsdam terms. Pinning their hopes on Konoe's not-yet-arranged mission to Moscow, the emperor and Kido [lord keeper of the privy seal, his closest advisor] waited and waited for a response from Moscow--a response that Ambassador Satô and others repeatedly stated would never come. Only after Hiroshima had been bombed did the emperor say, "We must bow to the inevitable;" now "is a good chance to end the war." More than ten thousand Japanese people died from conventional air raids during this eleven-day interval.SOURCE: Hirohito and the Making of Modern Japan, by Herbert P. Bix (HarperCollins, 2000), pp. 521-522
The Japanese "peace" overtures to the Soviets, which had followed Germany's capitulation, were vague, feeble, and counterproductive. They certainly never constituted a serious attempt to negotiate an end to the war. The thinking behind those maneuvers never progressed beyond decisions reached by the inner cabinet in mid-May 1945. As Konoe rightly suspected it would, the emperor's attempt to end the war via Moscow turned out to be a complete waste of time, and amounted to an imperial decision to postpone facing reality.
Did the U.S. also miss an opportunity by continuing to insist on unconditional surrender? Here's what Bix has to say (p. 518).
If the conservative [former ambassador to Japan] Joseph C. Grew and the "Japan crowd" had gotten their way and the principle of unconditional surrender had been modified in advance, it is highly unlikely that Japan's postsurrender leaders, now the "moderates" around the throne, would ever have discarded the Meiji constitution and democratized their political institutions. Grew and those who took his position had very little understanding of the Japanese body politic, no faith in the capacity for democracy of ordinary Japanese people, and certainly no desire whatsoever to see the social foundations of the monarchy dismantled.