Mr Putin “stressed that the (Russia-US) relations are the most important factor for providing strategic stability and international security”, according to a Kremlin statement on Sunday.
“He confirmed that Russia is open for dialogue with the USA on the most wide-ranging agenda.”
It follows a series of failed attempts to hold a new summit between the two leaders, including a planned meeting at November’s G20 in Argentina that Mr Trump abruptly cancelled, citing tensions in Ukraine.
Mr Trump and Mr Putin also failed to hold a fully-fledged meeting in Paris that same month on the sidelines of the centenary commemoration of the Armistice.
The two leaders held their one and only summit in Helsinki in July, which was widely considered to have been disastrous for Mr Trump and his reputation at home and abroad.
Moscow has said one of the key issues it wanted to discuss with the US is Washington’s plans to withdraw from a Cold War-era nuclear arms pact.
Russian foreign minister Sergei Lavrov was quoted as saying it was now up to the US whether to hold a new meeting in 2019.
“The issue should be addressed to Washington. Both our president and his representatives have said that we are ready for the talks when Washington is ready for it,” TASS news agency quoted Mr Lavrov as saying in televised remarks.
In a separate letter to Syrian president Bashar al-Assad, Mr Putin pledged continuation of aid to the Syrian government and people in the “fight against terrorism, in defence of state sovereignty and territorial integrity”.
Mr Putin also sent new year greetings to other world leaders including Theresa May and Japan’s Shinzo Abe, as well as Chinese president Xi Jinping.
Mr Putin wished “wellbeing and prosperity to the British people”, the Kremlin said.
Russia’s embassy in London said on Friday Moscow and London had agreed to return some staff to their respective embassies after they expelled dozens of diplomats early this year.
Britain expelled 23 Russian diplomats over accusations the Kremlin was behind a nerve toxin attack in March on former double agent Sergei Skripal and his daughter in the English city of Salisbury.
Russia, which denies any involvement in the poisoning, sent home the same number of British embassy workers in retaliation.
Additional reporting by Reuters
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