Putin's spokesman, Dmitry Peskov, said that Velayati handed Putin letters from Khamenei and Iranian President Hassan Rouhani.
The statement from Ali Akbar Velayati, a senior adviser to Iran's Supreme Leader Ayatollah Ali Khamenei, came in the wake of his meeting in Moscow with Russian President Vladimir Putin.
A senior advisor to Iran's Leader is in Moscow to hold talks with top Russian officials, including President Vladimir Putin. Washington says it will reimpose sanctions and has told other countries to halt all imports of Iranian oil from November 4 or face U.S. financial measures.
Iran's Leadership has long viewed relations with Moscow as "strategic", said Velayati, adding that the Islamic Republic "pursues this policy with the Leader's guidelines".
Velayati, who is also expected to visit China in the near future, described Trump as an "unreliable individual... with relation to global law who necessitates more and more cooperation (between Iran and Russia)".
An Iranian delegation led by Ali Akbar Velayati met Putin in the Russian capital on July 12, the Kremlin said.
Mr. Putin's reports in the meeting demonstrated that Iran-Russia oil cooperation can be developed to some $50 billion which is a considerable volume, he said, adding that Russia can appropriately replace Western companies which have left Iran due to United States sanctions.
Putin's meeting with Velayati followed Wednesday's talks in Moscow with Israeli Prime Minister Benjamin Netanyahu, who restated Israel's determination that the Iranian military presence be removed from Syria entirely.
Apart from the United States, the other signatories to the 2015 Iran nuclear deal - Russia, China, Germany, France and Britain - have said they still support the accord.
But European firms are reluctant to risk far-reaching U.S. financial penalties they would face for doing business in Iran.
He added that the cooperation in the region between the resistance front led by Iran and Russian Federation against terrorism and its supporters in Syria and other regional countries was indicative of an exemplary, strategic and long-term partnership between the two countries.
Velayati reaffirmed Iran's firm intention to maintain its presence in Syria, but skirted a question about a possible pullback from the border, saying only that Tehran won't bow to USA and Israeli coercion.
Lebanon's Shia Hezbollah group is accused of carrying out the bombing of the Jewish centre and of an attack on Israel's embassy in Buenos Aires two years earlier at Iran's demand.