The Russian army's chief of staff General Valery Gerasimov sent the letter to Dunford on July 19, saying Moscow was willing to cooperate with Washington in demining the Arab country and helping refugees return to their homes.
Reuters reported the letter on Friday, citing a US government memo.
"It is disappointing that the American side is unable to comply with the agreement on publicizing the content of contact only after both sides agree," Russia's Defense Ministry said in a statement on Saturday, Presstv reported.
In the letter, Gerasimov said Moscow was ready to “work with the Syrian authorities to provide security guarantees to refugees in the Rukban camp in the US-controlled Al-Tanf area and to create conditions for their return to their homes."
Rukban situated within a 55 kilometers so-called de-confliction zone set up by the United States to ensure the safety of its military base close to the Iraqi-Syrian border.
"A proposal was also made to coordinate humanitarian de-mining, including in Raqqa, and other priority humanitarian issues," the Russian ministry said.
Addressing the UN Security Council last week, Russian Deputy Ambassador Dmitry Polyansky urged world powers to help Syria's economic recovery and the return of refugees as Damascus nears final victory over terrorist groups.
Polyansky said the "revival of the Syrian economy" was a "critical challenge," with Syria facing an acute shortage of construction material, heavy equipment and fuel to rebuild areas destroyed by war.
In July, Moscow also sent a proposal to Washington to jointly organize the return home of Syrian refugees from Jordan, Turkey, Lebanon and Egypt that would involve international financial support.
Russia has military advisers in Syria and runs a number of military bases, from which it has been carrying out an aerial bombardment campaign against Daesh Takfiri militants on behalf of the Syrian government since September 2015.