According to a translation by Mr Trump's team the letter was dated Dec 15 at the Kremlin, addressed to "His Excellency Donald Trump" and signed "V. Putin".
In the four paragraph missive Mr Putin spoke of the importance of "ensuring security and stability in the modern world" and "bringing "bilateral co-operation to a qualitatively new level".
The Russian leader also wished Mr Trump and his family "warmest Christmas and New Year greetings" and "sound health, happiness, well-being, success and all the best".
A spokesman for Mr Putin confirmed the Russian president had sent Mr Trump a Christmas letter.
Mr Trump said: "A very nice letter from Vladimir Putin. His thoughts are so correct."
But in what some viewed as a veiled warning Mr Trump added " I hope both sides are able to live up to these thoughts, and we do not have to travel an alternate path."
The president-elect's remark about a new arms race was made on Friday in an off-camera phone interview with Mika Brzezinski, a well-known US television journalist, who reported it immediately.
"We can say with certainty - we are stronger now than any potential aggressor. Anyone!"Vladimir Putin
She had asked Mr Trump to clarify a statement he made on Twitter the previous day. In that statement he had said the US "must greatly strengthen and expand its nuclear capability until such time as the world comes to its senses regarding nukes".
Mr Trump's spokesman Sean Spicer later said the president-elect's intention was to put other countries on notice not to expand their nuclear capabilities.
Mr Spicer said: "It's a warning to them that this president isn't going to sit idly by. He's a man of action. He's not going to sit back and allow that to happen without acting in kind.
"He will not take anything off the table. He is going to do what it takes to protect our country. And what's going to happen is they will come to their senses and we will all be just fine."
At the same time Mr Trump was declaring an arms race Mr Putin was holding his end of year press conference in Moscow.
In it Mr Putin said he was unconcerned about the prospect of a US nuclear build-up, and called Mr Trump's tweet "nothing surprising".
Mr Putin said: "In the course of his election campaign he spoke about the necessity of strengthening the US nuclear arsenal, and strengthening the armed forces. There's nothing unusual here."
He added: "If anyone is unleashing an arms race it's not us. We will never spend resources on an arms race that we can't afford."
In conciliatory remarks Mr Putin also praised the US president-elect for "uniting society" and listening to the will of the American people.
He added: "No one thought he could win except us." Mr Putin said he would be prepared to visit Mr Trump in Washington if invited.
He said: "If Trump invites me to travel to the United States I will of course go. I expect a change in our relations.
They can't be worse."
Mr Putin also accepted the US military, not Russia's, was the most powerful in the world.
He said: "Nobody is arguing with that."
Joseph Cirincione, president of the Ploughshares Fund, which works to prevent the spread of nuclear weapons, said: "Trump is sewing chaos through our alliances. The whole world is listening to this. China is listening very closely to this. Do they need more weapons? What about Pakistan and India?
"Ronald Reagan never said something like this. In nuclear policy every word matters. This is unprecedented and bizarre."
This article was originally published on The Telegraph
