Meeting his Georgian counterpart ahead of an informal summit of leaders of the post-Soviet CIS states near St. Petersburg, Russia's Dmitry Medvedev said: "I believe we are capable of resolving all problems" without mediators.
Mikheil Saakashvili said there were a lot of "unresolved issues in bilateral relations, but none were unsolvable."
Tensions between the former Soviet republics have been strained since the Western-leaning Saakashvili came to power in the Caucasus country in early 2004.
The long-running row over Moscow's alleged support for Georgia's breakaway regions, along with Tbilisi's plans to join NATO, have been major factors behind the ongoing dispute. In the latest development, Georgia accused Russia of shooting down an unmanned reconnaissance plane over Abkhazia in April. Moscow denies any involvement in the incident.
Saakashvili said the "current situation was unnatural and of no benefit to any state."
It was Medvedev's first meeting with the Georgian leader since he took office in early May replacing Vladimir Putin, under who relations between the countries plunged to an all-time low.
Earlier on Friday, Medvedev met with Moldovan President Vladimir Voronin and Ukrainian President Viktor Yushchenko at the sumptuous Constantine Palace on the Gulf of Finland near St. Petersburg.
