The two presidents are in Ashgabat, the capital of Turkmenistan, to attend an international forum with other Central Asian leaders also in attendance on Friday.
This is the first meeting of the two presidents.
According to Ria Novosti, the Russian president told his Iranian counterpart that the two countries share views on international events.
Moscow and Tehran actively cooperate on the international stage and often agree on their assessments of world events, Putin said.
Putin invited the Iranian president to pay an official visit to Russia. Pezeshkian gratefully accepted the invitation.
According to the Iranian media, Pezeshkian, for his part, said that "Our relationship with Russia is sincere and strategic."
"Economically and culturally, our relations are getting stronger and stronger as time passes. The growing trend of cooperation between Iran and Russia should be accelerated in accordance with the will of the top leaders of the two countries and the necessity of strengthening these relations," the Iranian president said.
Stating that Iran and Russia have good mutual and complementary potentials and can help each other, Pezeshkian added that, "Our stances in the world are much closer to each other than any other states."
Expressing hope for speeding up the signing of the strategic partnership document between Iran and Russia, the Iranian president emphasized the development of bilateral cooperation. He added that, "I hope we will finalize this agreement during the upcoming BRICS summit in Russia."
Pezeshkian went on to point to the developments in Palestine and Lebanon and the Zionist Israeli regime's war on them, saying that "The Zionist regime does not respect any international legal and humanitarian framework and the situation in the region is dire; European and American countries do not want the relations between the countries of the region to continue calmly."
MNA