US President-elect Donald Trump's transition team have released a letter that they say was sent to him by Russian President Vladimir Putin.
"A very nice letter from Vladimir Putin; his thoughts are so correct," Mr Trump said about the note, which is dated 15 December 2016.
On Thursday the two leaders called for their respective nations to boost their nuclear arsenals.
Earlier, Mr Trump seemed to welcome the notion of a nuclear arms race.
"Let it be an arms race because we will outmatch them at every pass and outlast them all," MSNBC journalist Mika Brzezinski says the President-elect told her in a statement over the phone, in response to a question about his tweet from the day earlier.
On Thursday Mr Trump tweeted that the US "must greatly strengthen and expand its nuclear capability", only after hours after President Putin had called for his own military to "strengthen the military potential of strategic nuclear forces".
In the letter released by the Trump transition team, Mr Putin says he hopes that "we will be able - by acting in a constructive and pragmatic manner - to take real steps to restore the framework of bilateral cooperation in different areas as well as bring our level of collaboration to the international scene to a qualitatively new level".
Experts believe that Mr Putin hopes that the next US president will remove economic sanctions by the US Department of Treasury which have been placed on Russian officials following the invasion and annexation of Crimea.
At an annual media briefing on Friday in Moscow, the Russian president said he saw nothing remarkable in Mr Trump's tweet, making it clear that he does not view the US as a potential aggressor.
Donald Trump has been seen as close to Mr Putin and the Russian government, and drew condemnation from both Republicans and Democrats when he announced his selection of Rex Tillerson to be secretary of state, the top US diplomat.
The CEO of ExxonMobil has worked closely with Russian state oil company Rosneft, spoken out against international sanctions imposed on Moscow, and in 2013 was awarded an Order of Friendship by the Kremlin.
In response to the note from Mr Putin, the Republican president-elect praised the Russian president's words, calling them "so correct".
"I hope both sides are able to live up to these thoughts, and we do not have to travel an alternate path," the New York billionaire concluded.