NATO also wants to start a "dialogue with Russia" after Mr Trump and Mr Putin say they will work at "constructive co-operation".
Donald Trump and Vladimir Putin have agreed they will work together to tackle international terrorism and extremism, the Kremlin says.
The two men spoke for the first time on the telephone since Mr Trump's unexpected election victory last Tuesday, with the Kremlin adding they will work towards "constructive co-operation" to improve "the extremely unsatisfactory state of Russian-US relations at present".
Moscow said the pair would continue contact by phone but would not be meeting before Mr Trump is inaugurated as President on 20 January next year.
A Kremlin spokesman said Syria and the Crimea were not among the topics discussed during the first call.
A possible thawing of East-West relations appeared to be having an effect when NATO also appeared to reach out by saying the military alliance wanted more discussions with Russia.
Secretary General Jens Stoltenberg said: "The message from NATO has been that we want dialogue with Russia.
"Russia is our biggest neighbour, Russia is there to stay and especially when tensions run high and especially when we face many different security challenges, it is important to have dialogue."
Mr Trump will enter the White House as relations between Barack Obama and Mr Putin have become strained over several issues including the Syrian war, where the countries back different sides.
There has also been tension in recent years over the Ukraine conflict, while allegations of Russian cyber attacks featured in the election campaign.
Mr Trump's team said he and Mr Putin discussed a range of issues including the threats and challenges facing the United States and Russia, strategic economic issues and the historical US-Russia relationship.
A statement added: "President-elect Trump noted to President Putin that he is very much looking forward to having a strong and enduring relationship with Russia and the people of Russia."
During the election race, Mr Trump had warm words for the Russian president, suggesting he was a stronger leader than Mr Obama.
When asked how he would be able to change Mr Putin's stance on key issues, Mr Trump said he thought he could.
He said: "I think I would have a very, very good relationship with Putin, and I think I would have a very, very good relationship with Russia."
After Mr Trump won the election, Mr Putin said Russia was ready to do its part to fully restore ties with Washington.
Mr Trump's apparent warmth to Mr Putin and his election campaign comments that NATO was "obsolete" had led some European nations to express concern about his foreign policy.
President Obama, who is on his last tour of Europe before Mr Trump takes office, assured people in his first stop Greece on Tuesday that the US considers the NATO alliance a cornerstone of its security.
Mr Stoltenberg added that he was adamant that the Ukraine's "territorial integrity" should never be violated and that he agreed with Mr Trump that European NATO members should step up their defence spending.
The UK Prime Minister Theresa May's spokeswoman said on Tuesday that Britain hopes to raise the issue of NATO with Mr Trump at any early engagement.
Donald Trump and Vladimir Putin have agreed they will work together to tackle international terrorism and extremism, the Kremlin says.
 |
| Donald Trump and Vladimir Putin have spoken on the phone |
The two men spoke for the first time on the telephone since Mr Trump's unexpected election victory last Tuesday, with the Kremlin adding they will work towards "constructive co-operation" to improve "the extremely unsatisfactory state of Russian-US relations at present".
Moscow said the pair would continue contact by phone but would not be meeting before Mr Trump is inaugurated as President on 20 January next year.
A Kremlin spokesman said Syria and the Crimea were not among the topics discussed during the first call.
A possible thawing of East-West relations appeared to be having an effect when NATO also appeared to reach out by saying the military alliance wanted more discussions with Russia.
Secretary General Jens Stoltenberg said: "The message from NATO has been that we want dialogue with Russia.
 |
| A mural on a restaurant wall in Lithuania showing Donald Trump and Vladimir Putin, which followed warm campaign comments by the former towards Russia |
"Russia is our biggest neighbour, Russia is there to stay and especially when tensions run high and especially when we face many different security challenges, it is important to have dialogue."
Mr Trump will enter the White House as relations between Barack Obama and Mr Putin have become strained over several issues including the Syrian war, where the countries back different sides.
There has also been tension in recent years over the Ukraine conflict, while allegations of Russian cyber attacks featured in the election campaign.
Mr Trump's team said he and Mr Putin discussed a range of issues including the threats and challenges facing the United States and Russia, strategic economic issues and the historical US-Russia relationship.
A statement added: "President-elect Trump noted to President Putin that he is very much looking forward to having a strong and enduring relationship with Russia and the people of Russia."
During the election race, Mr Trump had warm words for the Russian president, suggesting he was a stronger leader than Mr Obama.
 |
| Relations between Vladimir Putin and Barack Obama have become strained |
When asked how he would be able to change Mr Putin's stance on key issues, Mr Trump said he thought he could.
He said: "I think I would have a very, very good relationship with Putin, and I think I would have a very, very good relationship with Russia."
After Mr Trump won the election, Mr Putin said Russia was ready to do its part to fully restore ties with Washington.
Mr Trump's apparent warmth to Mr Putin and his election campaign comments that NATO was "obsolete" had led some European nations to express concern about his foreign policy.
 |
| Relations between Vladimir Putin and Barack Obama have become strained |
President Obama, who is on his last tour of Europe before Mr Trump takes office, assured people in his first stop Greece on Tuesday that the US considers the NATO alliance a cornerstone of its security.
Mr Stoltenberg added that he was adamant that the Ukraine's "territorial integrity" should never be violated and that he agreed with Mr Trump that European NATO members should step up their defence spending.
The UK Prime Minister Theresa May's spokeswoman said on Tuesday that Britain hopes to raise the issue of NATO with Mr Trump at any early engagement.
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