Medvedev was inaugurated as Russia's third president some two weeks ago, taking over from Vladimir Putin, who was approved as the country's premier by parliament on May 8.
During the visit, Kouchner and France's secretary of state for European affairs, Jean-Pierre Jouyet, are also scheduled to meet with the Russian foreign minister, Sergei Lavrov.
The visit comes just five weeks before France assumes the EU presidency, which will last until December this year.
"Russia is a major EU partner. Taking into account our cooperation, the main aim [of the visit] will be the mutual development of long-term stable relations," the French Foreign Ministry said in an official statement.
In Moscow, the outspoken left-leaning Kouchner, a former French health minister and UN governor of the Serb province of Kosovo, will focus on the priorities of the Russia-EU partnership, especially in energy, migration and European security policy.
The French foreign minister is also set to discuss topical international issues and Russia's regional policy in Georgia and Ukraine.