A telegram was sent to to the US President from Putin as one of a series of messages issued to world leaders on Saturday, the Kremlin said.
Putin said: “The development of a constructive Russian-US dialogue is particularly important for strengthening strategic stability in the world and finding the optimal answers to global threats and challenges.”
The Russian President also added his hopes for “equality and mutual respect”.
It comes as tensions between the two nations have been high and some commentators have even described it as a “New Cold War”.
Russia has been accused of tampering in democratic elections in the West, including the 2016 US Presidential election and the Brexit referendum.
Putin also messaged the Syrian dictator Bashar al-Assad to say Russian will not abandon him in the New Year.
He said: “Russia will continue to render every assistance to Syria in the protection of state sovereignty, unity and territorial integrity.”
It comes as Putin ordered Russian forces to start withdrawing from the region this month.
But, he added that Russia will hold on to its Syria-based naval facility in Tartous and its air base in Hmeymim.
Russia’s intervention in Syria in 2015 preserved Assad’s regime despite international calls for the despot to be removed for his human rights abuses.
Putin also messaged other heads of state including Germany’s Angela Merkel, France’s Emmanuel Macron, and the leaders of former Soviet countries.
Putin sent a telegram to the US President in November 2016 saying he hopes Russia and the US can rebuild their relationship which was in “crisis”.
And as a follow up during a television address, the Russian President congratulated the tycoon on his victory in the race to the White House and said had followed the election closely.
Putin spoke of building a “constructive dialogue” between Moscow and Washington based on principles of equality and “mutual respect” in the interests of their two countries and of the wider world.
In a statement, the Kremlin said: “Putin expressed hope for joint work to restore Russian-American relations from their state of crisis, and also to address pressing international issues and search for effective responses to challenges concerning global security.”
But Trump has been under fire at home amid allegations of collusion between him and Putin during the 2016 US Election.
Earlier this month, Trump said there was “absolutely no collusion” between his presidential campaign and Moscow after a former adviser admitted lying to the FBI about contacts with Russia.
In his first comment on a guilty plea by his first national security adviser Michael Flynn about lying to the FBI, the US President told reporters there was no Russian meddling in his campaign which resulted in his victory.
Speaking to reporters as he departed the White House, he said: “What has been shown is no collusion, no collusion. There’s been absolutely no collusion.”
Putin condemned Donald Trump’s opponents for using “fabricated” stories about Russian collusion during the 2016 election as a “weapon” against him.