It is hoped the landmark call between the two Cold War rivals will help repairs relations which had deteriorated under Barack Obama’s administration.
Trump has vowed to improve ties with Russia and suggested he may lift the sanctions imposed on Moscow if Putin supports the US in the fight against terrorism.
Both leaders will be hoping for better relations between the two countries
Speaking to the Wall Street Journal earlier this month, Trump said: “If you get along and if Russia is really helping us, why would anybody have sanctions if somebody’s doing some really great things?”
Putin is believed to be eager to turn over a new leaf and congratulated Trump in his New Year message, saying he hopes Russia and the US would be able to "take their interaction in the international arena to a whole new level."
Theresa May is set to meet President Trump later today
The Russian leader added: "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.”
Russian Foreign Minister Sergey Lavrov called for the "re-establishment of normal relations" between the two countries now that Trump is in the White House.
Lavrov said: "Following the difficult relations we had under Barack Obama, President Putin is ready to meet in the interests of global security and stability.”
President Trump's Key Moments
Wed, January 25, 2017
Our pick of the top key moments so far for the 45th President of the United States, Donald J. Trump.
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US President Donald Trump signs an executive order to start the Mexico border wall project at the Department of Homeland Security facility
If you get along and if Russia is really helping us, why would anybody have sanctions if somebody’s doing some really great things?
Obama’s administration was very unpopular with Russian ministers who repeatedly criticised the former US President.
Russian prime minister Dmitry Medvedev wrote in a Facebook post ahead of Trump’s inauguration: “The Obama administration has destroyed relations between the United States and Russia, which are at their lowest point in decades.
Trump wants to work with Putin to tackle global terrorism
"This is its key foreign policy mistake which will be remembered by history.
"It is important to remember that Russia-US relations, without exaggeration, determine the fate of major international initiatives.”
British Prime Minister Theresa May, who has arrived in Washington ahead of a meeting later today, warned Trump to be cautious when dealing with the Russian president.
She said: "With President Putin, my advice is to engage, but beware.
"We should engage with Russia from a position of strength and build the relationship, systems, and processes that make cooperation more likely than conflict."