There was no high-level discussion about dropping of a second bomb. In fact, to my knowledge there is no evidence from the period that indicates Truman was aware a second bomb was to be dropped so soon after the first.

First, it is key to know that Truman himself was not tightly "in the loop" regarding the targeting of the atomic bombs. He was consulted and informed only at a few junctures. The idea that there was a very deliberate "decision" on his part is not believed by any present-day historians. Truman was much more on the margins about these things than most people realize.

He did, in late July 1945, ask about the "schedule" about the atomic bombings. The answer he got, as far as the documentary record reflects, was exclusively about the "tested type" of bomb, the implosion design. He was told there would be one ready around Augut 1st and another around the 24th. Truman's journal entries do not refer to the fact that there was another type of bomb in the works as well (the uranium bomb). He no doubt was told at some point that they were pursuing two types of bombs but it is not clear he internalized this regarding the schedule. The bombing order, which Truman was shown, gave very specific instructions regarding the first bomb but also authorized further bombing as weapons were available. I have seen no evidence that Truman understood how blanket an order this was until after Nagasaki.

In any cased, even the military schedule got modified due to factors on the island of Tinian. The initial use of the bomb was authorized for anytime on or after August 3rd. Weather conditions over Japan pushed it back to August 6th. The second bomb was scheduled for August 10th; due to weather it was moved up to Augusts 9th. So instead of a full week separating the two attacks, it was only 3 days. These decisions were made by those in Tinian, not Washington.

I have been working on a paper about Truman's actions in the days between the Hiroshima bombing and his order, on August 10th, that all atomic bombing must stop without explicit authorization by him. The argument, as I am sure you can intuit here, is that Truman did not really understand what was going on, and that after Nagasaki he sought to reassert control of the situation. In any case, there is no source that indicates that Truman explicitly authorized the second bombing. What we have regarding Truman and his advisors in this period indicate they talked about many other things (like the consequences of the Potsdam meeting, from which Truman just returned).

I have written a bit about these issues here and here. On the Washington/Tinian divide, in terms of actual control and decision-making, Michael Gordin's Five Days in August has a lot about this.