"We are holding talks with Russia's minister for transport. We would like to...start work in the near future - in the next month," said Vyacheslav Eshba.
Russia recognized both Abkhazia and South Ossetia as independent states on Tuesday.
Eshba said initial flights would be to Moscow and the southern Russian cities of Rostov-on-Don and Krasnodar.
He also said that the airport could eventually see flights to other countries, including Turkey, where many Abkhazians live.
Eshba also added that Abkhazia was also planning to eventually launch its own national air carrier.