Kremlin spokesman Dmitry Peskov said in a conference call with reporters that Moscow saw things differently.
"Russia has friendly partner-like relations with Iran, we cooperate on a wide range of issues, value our trade ties, and hope to develop them further," said Peskov, Reuters reported.
He was responding to comments Trump made to Fox News in an interview aired at the weekend in which he complained that Iran had "total disregard" for the United States.
Peskov said there was no reason for policy differences over Iran to hinder such a rapprochement.
"It's no secret for anyone that Moscow and Washington hold diametrically opposed views on many international issues," said Peskov. "That should not be an obstacle when it comes to forging normal communication and pragmatic mutually-beneficial relations between Russia and the United States."
Trump and Putin say they want to try to rebuild US-Russia ties that were badly damaged after the 2014 reunification of Crimea with Russia and Western sanctions imposed in response.
Russia criticized the Trump administration's move on Friday to impose sanctions on Iran after a recent ballistic missile test, saying the test did not violate existing agreements.
"We are sorry that things are working out in this way," Sergei Ryabkov, Russia's Deputy Foreign Minister, said on Monday.
China also on Monday said it had "lodged representations" with the United States over its new sanctions list targeting Iran, which includes Chinese companies and individuals.
Chinese Foreign Ministry spokesman Lu Kang said that Beijing had lodged a protest with Washington, and that such sanctions, particularly when they harmed the interests of a third party, were "not helpful" in promoting mutual trust.
"We have consistently opposed any unilateral sanctions," Lu told a regular press briefing.
The sanctions on 25 people and entities came two days after the Trump administration put Iran “on notice”.
Those affected by the sanctions cannot access the US financial system or deal with US companies, and are subject to secondary sanctions, meaning foreign companies and individuals are prohibited from dealing with them or risk being blacklisted by the United States.
The list includes two Chinese companies and three Chinese people, only one of whom the US Treasury Department explicitly said was a Chinese citizen, a person called Qin Xianhua.
Executives of two Chinese companies included on the list said on Sunday they had only exported "normal" goods to the Middle Eastern country and didn't consider they had done anything wrong.
China has in the past been angered by unilateral sanctions placed on Chinese firms by the United States and others.
China has close economic and diplomatic ties with Tehran, but was also instrumental in reaching the Iran nuclear deal in 2015.