But instead, the US and its allies forged ahead with NATO expansion, which destroyed Russia's faith in a less antagonistic future with its former foes.
"They do not think we respect them," Graham said.
Jack Matlock, a former US ambassador to Moscow, agrees. Matlock was in Russia during the chaotic period when the Soviet Union was dissolving. Like Bush's former special assistant, Matlock thinks the US could have done a better job at creating a connection with Moscow.
The expansion of NATO under Presidents Bill Clinton and George W. Bush were the equivalent of "a swift kick to the groin" for Russia, he wrote in The Washington Post last month.
"The sad fact is that the cycle of dismissive actions by the United States met by overreactions by Russia has so poisoned the relationship that the sort of quiet diplomacy used to end the Cold War was impossible when the crisis in Ukraine burst upon the world's consciousness," Matlock wrote.
This is what is making the present situation so dangerous, Graham says.
"How do you de-escalate at this point?" he asked. "We do not have the necessary channels of communication."
This was painfully obvious last week in Geneva, when all sides signed on to a compromise that none was really prepared to back.
The agreement called for "all illegal armed groups" to lay down their weapons. Almost immediately disputes arose over who was illegal, who was to go first, and who was responsible for ensuring compliance.
"All the parties were convinced that the other side was controlling the situation," Graham said.
The US State Department is endorsing photographs it says prove that Russian forces are instigating a separatist movement in eastern Ukraine. Russia, for its part, points to visits to Kyiv from the head of the CIA and the US vice president as evidence that Washington is "running the show" in Ukraine.
Tensions are so high at this point that anything could happen.
"We do not know how this is going to play out," Graham said. "No one really controls the forces. If there is widespread violence in eastern Ukraine, Putin can't not send forces in. But if troops move across the border, what will the West's response be? Sanctions aren't going to do it."
US Secretary of State John Kerry is conferring with European allies about imposing stiffer penalties on sectors of the Russian economy.
This might have some effect. Standard & Poor's ratings agency has already downgraded Russia's credit to just a step above junk.
But neither Washington nor Europe is prepared to sacrifice its own interests to penalize Russia.
"Right now they are desperately looking for sanctions that make them appear to be doing something but don't really cost them anything," Graham said.
If armed conflict breaks out between Russia and Ukraine, however, the stakes would rise in a hurry.
"We may have to provide lethal aid to Ukrainian troops," Graham said. "It is not difficult to spin out a scenario where things could get really nasty."
Washington has not really developed a suitable response to Russia in the Ukrainian crisis, he insists.
"We don't know what to do," he said. "Russia has outplayed us. We have been caught flat-footed, without a serious policy."
This will take a long time to play out, he maintains.
"It is definitely a different period in US-Russia relations," he said. "No one talks about a ‘reset' any more."
Back in 2009, then Secretary of State Hillary Clinton and her Russian counterpart staged a ceremony where they pressed a red button labeled "peregruzka," which the US thought meant "reset."
In fact, it means "overload," a mistake that amused Moscow and embarrassed Washington. But maybe overload is exactly right.
"Clinton should say that it wasn't really a mistake," Graham laughed. "It was a prediction."
