MOSCOW, Jan 28 (Reuters) - Russian President Vladimir Putin and U.S. President Donald Trump agreed to try to rebuild U.S. Russia ties and to cooperate in Syria, the Kremlin said on Saturday, after the two men spoke for the first time since Trump’s inauguration.

U.S.-Russia relations hit a post-Cold War low under the Barack Obama administration and Trump has made clear he wants a rapprochement with Moscow if he can get along with Putin, who says he is also keen to mend ties.

“Both sides demonstrated a mood for active, joint work on stabilizing and developing Russian-American cooperation,” the Kremlin said in a statement, adding that Putin and Trump had agreed to work on setting up their first meeting.

“The chat took place in a positive and business-like tone.”

Trump’s stance on Russia has been under intense scrutiny from critics who say he was elected with help from Russian intelligence agencies, a charge he denies. His detractors have also accused him of being too eager to make an ally of Putin.

For Putin, who faces possible re-election next year, an easing of U.S. sanctions imposed on Moscow over its role in the Ukraine crisis would be a major coup.

But the Kremlin made no mention of the subject being discussed, referring only to the two underlining the importance of restoring mutually-beneficial trade and economic ties.

Trump said on Friday he was only in the early stages of considering whether to lift the sanctions, as British Prime Minister Theresa May, other foreign officials and U.S. lawmakers cautioned that such a move would be premature.

The most tangible outcome of the phone call, as the Kremlin described it, appeared to be what it said was an understanding that jointly fighting international terrorism was a priority and that the two nations should cooperate in Syria.

UKRAINE, IRAN, KOREA

“The presidents spoke in favor of setting up genuine coordination between Russian and American actions with the aim of destroying Islamic State and other terrorist groups in Syria,” the Kremlin said.

That could signal a major policy change as, for now, cooperation is largely limited to coordinating to ensure that the two countries’ air forces operate safely and that the risk of accidental confrontation or collision is minimized.

Moscow is one of Syrian President Bashar Al-Assad’s allies, while Washington, under Obama, called for him to step down and backed rebel groups fighting to topple him.

The Kremlin said Trump and Putin had agreed to establish “partner-like cooperation” when it came to other global issues such as Ukraine, Iran’s nuclear program, tensions on the Korean peninsula and the Israeli-Arab conflict.

They had also agreed to stay in regular contact and had both said they wanted each other’s nation to flourish, the Kremlin said.

“He (Putin) reminded (Trump) that our country has supported America for more than two centuries, was its ally in two world wars and now views the United States as its most important partner in the fight against international terrorism,” the Kremlin said.

Since Russia annexed Ukraine’s Crimea in 2014, Kremlin-backed media have spent much of their time criticizing the United States and accusing it of trying to undermine Moscow. Since Trump’s election, it has backed away from that line.

(Reporting by Alexander Winning; Editing by Andrew Osborn)