The ongoing fighting, which broke out on Sunday, is the most violent in several years with artillery shelling, drones, tanks and helicopters.
Both Armenia and Azerbaijan have mobilized their military forces.
The Armenian-led breakaway region of Nagorno-Karabakh said on Monday that 32 people had been killed and hundreds injured in fighting with Azerbaijani troops on Sunday.
Another 28 people are said to have been killed on Monday.
Both sides accuse each other of killing civilians.
According to Azerbaijan, six Azeri civilians have been killed and 19 injured, reports Reuters.
Azerbaijan and the government of Nagorno-Karabakh both say they have taken important positions and that they have made gains.
Armenia: Turkey involved
On Monday, Armenia's ambassador to Russia said that Turkey, which was openly on Azerbaijan's side in the conflict, had sent 4,000 troops from northern Syria to Azerbaijan, reports the Russian news agency Interfax.
According to the ambassador, the Turkish troops are taking part in the fighting against Armenia.
The claim is denied by Azerbaijan, which calls it "complete nonsense", reports Reuters.
Encourages calm
The UN, the United States, China and Russia have called on the warring parties to calm down.
- We have good relations in the area.
We will see if we can try to stop this, said President Donald Trump on Sunday.
The conflict over Nagorno-Karabakh has been going on since the fall of the Soviet Union when the region, ruled by ethnic Armenians, broke away from Azerbaijan.
The area serves as an important corridor for oil pipelines from the Caspian Sea.