He was picked up by security forces searching the remote Pankisi Gorge near the border with Russia's war-torn Chechnya region.
State Security Minister Valeri Khaburdzania said the man, an "ethnic Arab", was carrying a French passport believed to be fake.
He was wearing military uniform under his clothes and carrying "Islamic literature" and a map of the gorge.
"In our opinion or, rather, according to our information, he is linked to certain terrorist organisations," the minister told Georgian television.
"We can certainly say that he is connected to organisations of the type which finally lead to al-Qaeda...
"As regards French citizenship, we believe that it is a forged passport."
Mr Khaburdzania said that no force had been used during the arrest and the man was now being interviewed.
He did not rule out extradition proceedings:
"We may have to extradite him because we believe that he is one of the prominent individuals linked to these terrorist organisations and he may have committed more serious crimes on the territory of other countries."
Anti-terror training
The security minister said he had his suspicions about an al-Qaeda connection because of the Pankisi Gorge's links to Khattab, the ethnic Arab commander of a Chechen militant group who was killed earlier this year.
Pankisi Gorge
Home to the Kists, a Muslim community with ethnic links to the Chechens
Local non-Muslims say they have been forced to move out by Chechen refugees
Chechen rebels have long used the gorge as a base
The US believes that al-Qaeda militants may also be active there
Washington has provided Georgia with military instructors to help combat the perceived threat from radical Islamic militants believed to be using the lawless gorge as a base.
News of the arrest came just hours after Georgian President Eduard Shevardnadze inaugurated a new phase in the US-funded anti-terrorist training programme.
Under the programme, four 500-man battalions are to be trained over two years at a cost of about $64m.
Georgia has rejected Russia's calls for permission to send its own forces into the Pankisi Gorge to pursue militants seeking refuge there.
Moscow, which is fighting Chechen separatist rebels just over the border, has been accused of mounting air strikes on the gorge, the latest of which, this month, left one civilian dead and seven wounded.
Georgia began a military sweep of the gorge shortly afterwards but reported no resistance.