This is actually a more complicated question than might first appear, so much so that there is basically an entire book on the subject of what was happening in the Soviet government between August and December 1991: Serhii Plokhy's The Last Empire: The Final Days of the Soviet Union.

On paper at least, from the time of the end of the August coup to December 25 Gorbachev was the Soviet President, in charge of the Soviet government (complete with its ministries of Justice, Defense, Internal Affairs, the KGB, etc). Even after all of the Soviet Socialist Republics had technically declared independence and the heads of Russia, Ukraine and Belarus had signed the Belovezha Accords on December 8 (which declared the USSR dissolved), Gorbachev was still active in office, receiving foreign dignitaries and granting interviews. Gorbachev, as Soviet President, had even co-chaired the Madrid Conference (dealing with peace proposals for the Middle East) on Oct. 30 - Nov. 1. This didn't really come to a head until the signing of the Alma-Ata Protocol by 11 of the 15 republics on December 21, when Russia was recognized as the legal successor to the USSR. It really was at this point that there was no more room for maneuver for Gorbachev, and he resigned as Soviet President (and turned over control of Soviet nuclear weapons), and the last vestiges of the Soviet government were folded into Yeltsin's Russian one.

Of course this ultimately ignores a lot of the facts on the ground from August 1991. The Communist Party of the Soviet Union had been banned in Russia at the end of August, and the Russian Presidency had seized its vast assets and properties. Therefore a major component of Soviet order had at a stroke been eliminated, and much of its tangible power transferred to the Russian republican government. Although Gorbachev remained head of the Soviet government, the coup plotters had come from the top of that very structure (they had mostly been Communist Party members who had been promoted and initially supportive of Gorbachev's reforms, but undertook the coup in order to stop the process). Gorbachev had himself been under house arrest until the collapse of the coup, at which point practical power lay with Russian President Yeltsin (Plokhy calls this the "counter-coup"). When Gorbachev returned to Moscow he was pretty publicly humiliated by Yeltsin, and forced to accept Yeltsin's appointees to run what was left of the Soviet government. The next four months would see a process of the Russian government (initially established as a separate, subordinate entity the year before) basically expand and absorb its Soviet-level counterparts within Russia proper, in no small part because the republics were withholding tax revenue from the union government.

Even before August, it had been clear that some kind of major constitutional reorganization of the USSR was going to happen. Gorbachev himself had led this through negotiations at the start of the year at Novo-Ogaryovo, and a new union treaty (to establish a "Union of Sovereign States" and replace the 1922 Soviet Treaty of Union) had been negotiated - the final formal signing ceremony of this was specifically what the coup plotters were trying to prevent by acting on August 19.

Once the coup of August 1991 failed, and the republics rushed in their formal declarations of independence, a continued union was plausible, but on ever-looser terms. The final nail in the coffin was the December 1 referendum in Ukraine on independence - it passed by an overwhelming margin (to Gorbachev's surprise), and also saw the election of Leonid Kravchuk to the post of Ukrainian President, established that same day. From this point, although Yeltsin himself would have preferred some sort of continued union, the Ukrainian government was pretty adamant on only the loosest possible connection to an international organization (which became the Commonwealth of Independent States, and even in this case Ukraine never formally became a member). Yeltsin basically did not want a continued union without Ukraine (as especially the Central Asian republics were deemed too reactionary and too much of an economic drain on Russia), and so at Belovezha and at Alma-Ata he accepted a dissolution of the USSR.

Long and short - could the Soviet Union as it existed before 1991 continued on in late 1991 or after? No, because fundamental political and constitutional changes were already underway and championed by Gorbachev. Could some sort of stronger union have taken its place? Possibly, and it's definitely something Gorbachev, Yeltsin, and a number of other republican leaders (like Kazakhstan's Nazarbayev) would clearly have preferred. But at the end of the day any such plans foundered because no one could agree who would be in charge of such a union, and with what powers, and the Ukrainian government basically vetoed any participation in such a union.