Interfax, a Russian news agency, quoted a representative of Sergei Bagapsh, Abkhazia's leader, as saying that two unmanned airborne drones were shot down on Sunday.
Georgia earlier alleged that a Russian MiG-29 military aircraft had shot down a Georgian spy aeroplane on April 20.
Tbilisi called that incident "an act of [Russian] aggression".
Relations between Georgia, a former Soviet region, and Russia have worsened in recent weeks.
Moscow recently said it was strengthening ties with Abkhazia and South Ossetia, another Georgian region which is looking to secede.
While Georgia wants to integrate with the West and in particular, the Nato military alliance, Moscow is concerned that Western influence is eroding Russian authority in its border regions.