President Ghani insists the peace talks should be "Afghan owned and led," according to the statement.
Lavrov said Russia agrees the peace process should be under the auspices of the Afghans and it is "ready to cooperate," the statement added.
Russia's Foreign Ministry, which had extended the invitation to the Taliban despite the group's official designation as a terrorist organization in Russia, also confirmed in a statement Monday that the talks have been postponed after Lavrov's conversation with Ghani.
″(Ghani) supported the idea of a Moscow meeting in principle and suggested postponing it in order to come up with a consolidated position," the ministry's statement said.