By Andrew Osborn and Maria Tsvetkova

MOSCOW (Reuters) - The Kremlin said on Monday it did not agree with U.S. President Donald Trump's assessment of Iran as "the number one terrorist state" and a Russian diplomat said any U.S. attempt to reopen an Iran nuclear deal would inflame tensions in the Middle East.

Trump and Vladimir Putin, his Russian counterpart, have made clear they want to try to mend U.S.-Russia ties, which have slid to a post-Cold War low in recent years. But starkly different approaches to Iran, as set out by a raft of top Russian officials on Monday, could complicate any rapprochement.

Their comments also suggest that a policy idea Trump and his aides are reported to be considering -- to try to drive a wedge between Moscow and Tehran -- may be a non-starter.

Trump told Fox News in an interview aired at the weekend that Iran had "total disregard" for the United States and labeled Tehran "the number one terrorist state", complaining it was sending arms and weapons "all over the world".

He spoke out after his administration put the Islamic Republic "on notice" following an Iranian ballistic missile test and imposed new economic sanctions on Friday.

Two sources told Reuters last week the Trump administration was now exploring how to renegotiate key terms of the 2015 accord between Tehran and six world powers, under which Iran agreed to curb its nuclear program.

But several top officials in Russia, which has worked closely with Iran to support President Bashar al-Assad in Syria, signaled on Monday that they fundamentally disagreed with Trump's approach to Tehran.

'OIL ON THE FLAMES'

When asked about Trump's description of Iran as a "terrorist state," Kremlin spokesman Dmitry Peskov told reporters that Moscow saw things very 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.

"It's no secret for anyone that Moscow and Washington hold diametrically opposed views on many international issues," he added, saying that should not hinder a rapprochement.

Russia's ambassador to Iran, Levan Djagaryan, said Moscow was concerned by escalating rhetoric between the United States and Iran, while Sergei Ryabkov, Russia's deputy foreign minister, said Washington's decision to impose new sanctions on Iran was a source of regret.

Ryabkov, in a separate interview with the Moscow-based Security Index Journal published on Monday, also urged Washington not to try to reopen the Iran nuclear deal, saying to do so would risk inflaming the Middle East.

"Don't try to fix what isn't broken," said Ryabkov. "It would be an undesirable and negative turn of events that would only serve to pour oil on the flames in the Middle East."

Trump has spoken of the possibility of cooperating with Russia to fight Islamic State.

In comments that further underlined how far apart Moscow and Washington are over Iran, Russian Foreign Minister Sergei Lavrov told reporters on Monday he thought Iran should be part of an international coalition to fight the militant group.

(Additional reporting by Denis Pinchuk; Editing by Catherine Evans)