Americans did invade part of the Japanese home islands such as Iwo Jima and Okinawa. The battles were atrociously brutal and horrendous and 94 % of Japanese forces were killed and Americans suffered enormous casualties. Because the resistance had been so fierce and ruthless, and because the Japanese military had declared that 20 million conscripts were ready to take arms to defend the Empire. Many American military officers wanted to just use the nuclear bombs and be done with it.
There was also the issue of the nascent Cold War; the Soviets wanted to invade Japan and turn it into a Communist, satellite country and Americans wanted the use of the two bombs to also be a show of force to make Staling reconsider.