The dispute between Azerbaijan and Armenia claimed 30,000 lives on both sides between 1988 and 1994 after the Azerbaijani region with a largely Armenian population proclaimed independence from Azerbaijan in a bid to join Armenia.
"There is hope here, but questions remain, and talks on this matter will be continued between Vladimir Putin and [Armenian leader] Robert Kocharyan," Sergei Lavrov told journalists.
He said that the two heads of state would attend a summit of the Collective Security Treaty Organization, which unites six former Soviet republics, in the Belarusian capital next week.
"Russia, as a participant of the OSCE Minsk Group [which also comprises the U.S. and France], is trying to contribute to achieving an agreement between the sides, but a settlement can be reached only on a mutually acceptable basis," Lavrov said.
The minister also said Putin and Azerbaijani leader Ilham Aliyev had common understanding of processes in the former Soviet Union.
Russia's defense minister, Sergei Ivanov, said May 31 that the conflict between Armenia and Azerbaijan should be resolved by the two countries themselves and not by third parties. He added peacekeepers could be deployed in the conflict zone.
The Russian minister's remarks came a day after Armenia and Azerbaijan both claimed fatalities following a shootout between soldiers on the border near Nagorny Karabakh and accused each other of breaching a 1994 ceasefire agreement.
More than 100 people have been killed in the region since the ceasefire was brokered.