They didn't come up with any kind of "position" in the intervening days. I think most people don't appreciate how tight of a schedule that was. I've gone into detail on this here. The short version is that the Japanese high command weren't willing to react to Hiroshima without making sure they understood what had happened there, so they sent a scientific survey team investigate. They didn't get results back until the evening of August 8th. That night, the Soviet Union invaded Manchuria. So they called a cabinet meeting the next morning to talk about both of these things. While they were in the meeting, a bomb was dropped on Nagasaki.

There is no tangible evidence that the bombing of Nagasaki had any real change in their opinions — they entered that meeting with the same views that they left that meeting. Which is to say, if you're looking for a "did they change their mind on Nagasaki" answer — there isn't one.

The Japanese also never stated any formal negotiating position. The US knew of Japanese goals and desires by tapping their communications abroad. But it isn't as fine-grained as a "day-by-day" change of their position. Even after Hiroshima, Nagasaki, and the Soviet invasion, the Japanese offered up what the US only considered to be a conditional surrender: they agreed to accept Potsdam as long as it didn't impinge on the status of the Emperor. The US rejected this as too conditional; the offer of unconditional surrender took place on August 14th, after an attempted coup by junior officers.