Trump has vowed to have better relations with Russia than his predecessor.
This month, he suggested he would lift sanctions imposed by the Obama administration
if the Kremlin helps the US battle terrorists.
In his new year's message, Putin congratulated Trump and expressed hope
their two countries would be able to "take their interaction in the international arena to a whole new level."
"Major global and regional challenges that our countries have confronted in recent years clearly confirm that Russia-US relations are an important factor in ensuring stability and security in the modern world," Putin said.
Russian Foreign Minister Sergey Lavrov this week called for the "re-establishment of normal relations" between Russia and the United States under the Trump administration.
"Following the difficult relations we had under Barack Obama, President Putin is ready to meet in the interests of global security and stability," Lavrov said.
Tensions between the US and Russia are at their highest in years, after the Obama administration accused the Kremlin of interfering in the 2016 election,
imposing sanctions and ejecting diplomats.
Trump has been accused by his critics of being too close to Russia, and both he and Obama were briefed on an unverified dossier
containing allegations that Russian operatives possessed compromising information on Trump.
Some 4,000 US soldiers have also been deployed to eastern Europe as part of troop rotations the Pentagon said are intended to bolster ties with NATO allies and send a message to Russia
There is therefore great scope for Trump to improve relations between Washington and the Kremlin, as he has expressed a desire to do.
This week, however, UK Prime Minister Theresa May advised caution when dealing with Russia, ahead of her meeting Friday with Trump.
"With President Putin, my advice is to engage, but beware," she said
"We should engage with Russia from a position of strength and build the relationship, systems, and processes that make cooperation more likely than conflict."