"Alexei Leonov was one of the greats in spaceflight. His early exploits in space set the stage for later accomplishments by both the Soviet Union and the United States. He will be missed by those who knew him, and NSS mourns his passing, extending condolences to the Russian spaceflight community", NSS Senior Operating Officer Bruce Pittman said in a statement.
The organisation added that Leonov was a pioneer and served as a great inspiration for future astronauts throughout the world.
European Space Agency head, Johann-Dietrich Worner, also expressed condolences over the famous Russian cosmonaut's passing in a letter sent to Dmitry Rogozin, head of Russia's space corporation Roscosmos.
"It is of great sadness that I learned of Alexei Leonov's passing. With his demise, one of the great heroes of space exploration has left us ... We all owe him a great deal. Please convey my sincere condolences to his family", Worner wrote.
On Friday, the Gagarin Cosmonaut Training Centre said that the legendary cosmonaut had passed away at age 85. Later that day, Russian President Vladimir Putin described the late cosmonaut as a hero and pioneer.
Leonov, a two-time Hero of the Soviet Union and the eleventh Soviet cosmonaut, made his 12-minute spacewalk on 18 March 1965, as part of the Voskhod 2 mission. In July 1975, Leonov participated in the first joint US-Soviet Union space mission. It involved the docking of the US Apollo and the Soviet Soyuz-19 spacecraft.
A memorial service and burial will take place in Russia's Federal Military Memorial Cemetery to the northeast of Moscow established to honour people who performed outstanding service to the nation in their lifetime, on 15 October.