But in recent days McCain has signaled that he intends to focus more on hard-line criticism of Russian Prime Minister Vladimir V. Putin and his government.
Obama too faces challenges as he tries to stake out a more conciliatory foreign policy that includes his willingness to negotiate with enemies.
His softer language toward Moscow leaves him open to criticism at a time when the U.S. is clearly aligned with Georgia, an emerging democracy. But Obama aides say this, in fact, can be an advantage.
"There is a fundamental difference, which is Sen. McCain wants to isolate Russia, and he thinks that is a way to change their behavior," said Stanford University professor Michael McFaul, an Obama advisor on Russia.
Aides to both candidates said Monday that each had been focused behind the scenes for months on the growing tensions between Georgia and Russia.
McCain, aides said, has been in regular contact for years with Georgian President Mikhail Saakashvili, and has traveled on several occasions to Georgia, even visiting the separatist region of South Osettia in 2006.
In April, after Putin forged stronger ties with separatist leaders, he called Saakashvili.
Obama advisors said that the Illinois senator had been involved, as well, saying in April that Georgia must "resist the temptation to be drawn into a military conflict."
As Obama traveled to Hawaii on Friday for a vacation, he issued a brief but bland statement calling on Georgia and Russia to "show restraint."
By Saturday he had spoken with the Georgian president and condemned Russia's "aggressive actions."
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peter.wallsten@latimes.com