The crisis between Georgia, a vital energy transit route, and its former Soviet master has alarmed Nato allies who see Georgia as a possible future member of the US-led alliance.
 
Abkhazia and South Ossetia broke from Tbilisi's control in the 1990s, seeking independence or absorption into Russia.
 
Thousands of people in the two areas have been issued with Russian passports.
 
Unanimous support
 
In his comments on Wednesday, Nato's Appathurai said: "The allies are unanimous in supporting, endorsing Georgia's territorial integrity and will not recognise or support steps that undermine that sovereignty."
 
He said that he was not aware that any troop movements had taken place by either side.
 
"Have no doubt, everyone is watching very carefully what is happening on the ground," he said.
 
Appathurai said that, under the mandate for peacekeeping operations in Abkhazia, plans for deployments needed to be approved by the conflicting sides.
 
Georgia has made it clear that it would not approve of further Russian deployments, he said.
 
The White House also expressed concern about Russia's troop build-up.
 
In a statement, Gordon Johndroe, the US national security council spokesman, said: "We're concerned about reports from the region. The president has been updated on the latest on the situation."

'War scenario'
 
For her part, the speaker of the Georgian parliament said on Wednesday that Russian plans to increase the number of Russian peacekeepers in Abkhazia was an attempt to provoke a "war scenario".
 
Nino Burdzhanadze, speaking from the Georgian capital Tbilisi, said that Russia was trying to involve Georgia in military action.
 
"Russia is trying to destabilise the situation in the country. The main goal of Russia is not to allow us to be a success story, not to allow us to continue our movement to the direction of Nato and to the direction of Europe."
 
Burdzhanadze said Russia had already begun strengthening its military presence in Abkhazia.