Nato foreign ministers have said they "cannot continue business as usual" with Russia, and demanded that Moscow pull troops from Georgia immediately.
The declaration followed talks in Brussels about the conflict between Moscow and Tbilisi over Georgia's breakaway region of South Ossetia.
Russia accused Nato of bias and of trying to save a "criminal regime".
Foreign Minister Sergei Lavrov said a withdrawal was possible within three or four days - if Georgia did likewise.
"The withdrawal will be determined by how effectively Georgia returns its forces to their permanent bases [held before the conflict]," Mr Lavrov said.
Some Russian troops have been seen leaving Gori, the largest Georgian town close to the South Ossetia border.
But BBC correspondents on the ground say there are still Russian artillery positions in place. In addition, there are Russian checkpoints about 35km (22 miles) from the capital, Tbilisi.
Georgia dismissed the move as a "show aimed at creating the illusion of a withdrawal".
The conflict broke out on 7 August when Georgia launched an assault to wrest back control of the Moscow-backed breakaway region of South Ossetia, triggering a counter-offensive by Russian troops who advanced beyond South Ossetia into Georgia's heartland.
A ceasefire was signed at the weekend, with Moscow pledging to begin pulling back its troops on Monday, but correspondents say there has so far been little sign of any large-scale withdrawal.
Both sides have accused the other of violating the EU-brokered peace plan.
Buffer zone
Following the crisis talks in Brussels, the 26 foreign ministers said in a joint statement that they could not have normal relations with Russia as long as Moscow had troops in Georgia.
PEACE PLAN
"The Alliance is considering seriously the implications of Russia's actions for the Nato-Russia relationship," the statement said, read out by Nato Secretary General Jaap de Hoop Scheffer.
"We have determined that we cannot continue with business as usual."
Mr de Hoop Scheffer added that he could not see how the Nato-Russia Council - set up in 2002 as a framework for dialogue - could convene at this time.
But he said lines of communication would not be abandoned.
"The future of our relations with Russia will depend on the concrete actions Russia will take to abide by the words of President Dmitry Medvedev [regarding the peace plan], which is not happening at the moment," Mr de Hoop Scheffer said.
He also said that the member states had agreed to set up a Nato-Georgia commission to strengthen ties with Tbilisi, but stopped short of giving a timetable for Georgia's accession to Nato.
The Russian military has warned that the withdrawal process will be slow until the weekend at least, and that troops will remain in an undefined buffer zone around South Ossetia.
It says such a move is permitted under the ceasefire deal which allows Russia to take additional security measures until international peacekeepers are deployed.
But Georgia says Moscow is going much further and that Russian troops have seized control of a key commercial port in Poti in an attempt to cripple the Georgian economy.
HAVE YOUR SAY
Earlier, in an apparent goodwill gesture Russia exchanged 15 Georgian prisoners for five of its own troops at a Russian checkpoint in Igoeti, about 30km (18 miles) from Georgia's capital.
Georgian officials told the BBC's Helen Fawkes, who was at the scene, that two of the Russian prisoners were airmen who had been shot down by Georgian forces about two weeks ago.
Meanwhile, the Organisation for Security and Co-operation in Europe (OSCE) said Russia and Georgia had agreed to allow 20 extra military observers to be deployed in and around South Ossetia.
In total, the OSCE said it would send up to 100 additional monitors to join the handful it already has in Georgia.
The OSCE has had a presence in South Ossetia since the end of a civil war there in the early 1990s, which resulted in de facto independence for the region.
It also supports a UN-led peace process in Georgia's other separatist region of Abkhazia.
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