A recent reply I gave to a similar question:

The US Strategic Bombing Survey, conducted in 1946, concluded that:

Based on a detailed investigation of all the facts, and supported by the testimony of the surviving Japanese leaders involved, it is the Survey's opinion that certainly prior to 31 December 1945, and in all probability prior to 1 November 1945, Japan would have surrendered even if the atomic bombs had not been dropped, even if Russia had not entered the war, and even if no invasion had been planned or contemplated.

There are other present-day sources which conclude the same thing, based on projections for Japanese food supplies (they were indeed quite isolated at that point due to blockade) and knowledge about the internal workings of Japan's high command.

If the surrender terms had been modified (e.g. from "unconditional surrender") it becomes almost a sure thing. If the Allies had given assurances about the postwar role of the Emperor it would have given those in the Japanese high command who favored a "diplomatic" solution a significant edge.

It is even today not clear that the atomic bombs were what caused the Japanese surrender; there is strong evidence that the invasion of Manchuria by the USSR is what precipitated the surrender in August 1945.

Anyway, there is obviously no way to know, but there is a rich literature on this question. Tsuyoshi Hasegawa's Racing the Enemy is a good place to start, as he lays out the scenarios very well. Anybody who tells you the alternatives were "drop two atomic bombs on cities in rapid succession without warning" versus "an invasion that kills millions of Americans" is trying to pull a historical "fast one" on you. The reality is a more nuanced and complex.

In general there were 3 broad options available to the US and allies in the Summer of 1945, as long as the Allies did not wish to revise the call for unconditional surrender.

1. Was drop the bombs and try to force a surrender ASAP, which obviously was the course chosen after an extensive series of talks between top brass in the Spring about an invasion.

2. Invasion, which was essentially decided on as the fall back if the bombs were not enough, with a tentative start date of November 1st, with additional bombs to be used in support. This would be commanded by MacArthur and he and Marshall were the main drivers and voices in Truman's ear.

3. Continue the blockade and bombardment of the Home Islands to starvation and subjugation. The islands by Summer 1945 were already cut off from the world for the most part. The Fast Carrier Force repeatedly conducted massive air raids over targets of opportunity, subs mined and patrolled the Sea of Japan, and even the cruisers and battleships moved in and shelled coastal targets. Continuation of this policy was the favored choice for the most part of the Navy, lead by Nimitz and King back in Washington.

Over the summer of 1945 there is evidence King didn't really consider the invasion going ahead a settled issue at all, especially as casualty estimates continued to rise, and intelligence rounded up the number of Japanese formations in place to defend the islands. And finally the Navy was still very wary of allowing MacArthur total control of the plan making him even more of a Generalissimo, the command structure had worked when Nimitz and mac could command separated demarked territories, but the invasion would have brought the fractured command in the Pacific to a head after having simmered since 1941.