Georgian forces have fired at an aircraft they believed to be Russian after it violated Georgian airspace, a senior government official said.
Tbilisi could not confirm whether the plane was shot down, but said that a nearby section of forest, in Abkhazia's Kodori Gorge, was on fire.
Russia dismissed the claim, and an air force spokesman called it "the latest provocation aimed against us".
The two ex-USSR states' relations have been strained since a 2006 spy row.
Earlier this month, Georgia said that two Russian planes had violated its border around the same area.
In a separate incident, Georgia also claimed a Russian plane had dropped a missile near the capital, Tblisi.
Russia strongly denied the accusations.
Georgian officials said Wednesday's incident was not reported earlier because they were still checking information.
The Russia defence ministry said Russian planes had not flown near the area on that day.
Georgian officials said the site where the plane may have crashed was in the Georgian-controlled territory of Upper Abkhazia, but that it was remote and difficult to reach.
Upper Abkhazia is the Georgian authorities' term for the strategic Kodori gorge, which is the only part of the breakaway republic of Abkhazia over which Georgia retains partial control.
Under an agreement between the two, Georgian defence forces and heavy weaponry cannot be deployed in the area. However, Georgian interior ministry forces can operate there.