Kremlin spokesmen said Gyurcsany has been invited to visit by Russian President Vladimir Putin.
During their upcoming talks in Moscow, the two leaders are expected to focus on energy cooperation, an issue that dominated the agenda of their previous meeting in the Russian resort city of Sochi last September.
Hungary, which is among Europe's most dependent countries on Russia for natural gas, has been increasingly concerned about its energy security since Moscow briefly halted supplies through Ukraine and Belarus over pricing rows in 2006 and 2007.
Last June, Russian gas monopoly Gazprom and Hungarian petroleum giant MOL Nyrt. signed an agreement on the possible extension into Hungary of a gas pipeline running under the Black Sea. This could make the Central European country a key hub for Russia's energy supplies across the EU.
Relations between the two Eastern Bloc countries have long been clouded by memories of the brutal Soviet suppression of the 1956 uprising in the Hungarian capital, Budapest.
During a visit to Hungary in March 2006 - the year that marked the 50th anniversary of the event - Putin acknowledged Moscow's moral responsibility.
In another conciliatory gesture, the Russian leader returned a collection of old books taken away by the Soviet Army as a war trophy during WWII.