Ukraine officials say the surge in fighting has killed at least 13 civilians and fighters on both sides since Sunday.
They say it's the worst violence since the two sides agreed to a new truce just before Christmas, on December 23.
The Organisation for Security and Cooperation in Europe (OSCE) is responsible for monitoring ceasefire violations and organising peace talks between envoys from Russia and Ukraine.
It also helped negotiate a February 2014 truce deal in Minsk.
The Organisation's Alexander Hug says the humanitarian situation in the town of Avdiyivka remains difficult.
"The town has no electricity, reduced and limited water supply, no or very limited heating, and of course it is located at the very close proximity of the fighting theatre that has emerged rather loud since the early hours in the morning of last Sunday."
Mr Hug has described the situation as unpredictable and unstable.
"And it is caused by the failure of the so-called DPR (Donetsk People's Republic) and the Ukraine armed forces to withdraw their heavy weapons. And it is caused by the close proximity of the armed formations and Ukraine armed forces."
NATO Secretary-General Jens Stoltenberg says the the ceasefire in eastern Ukraine was violated more than 5,600 times in recent days, leaving some 20,000 people without electricity amid freezing temperatures.
Mr Stoltenbeg called on Russia to intervene to help bring an end to the violence.
"We call for an immediate return to the ceasefire and withdrawal of all heavy weapons banned under the Minsk agreements. OSCE monitors must be guaranteed free and safe access throughout the region. This must include Ukraine's border with Russia. We call on Russia to use its considerable influence over separatists to bring the violence to an end. All parties should implement the commitments under the Minsk agreements in full."
Ukraine and NATO accuse Russia of supporting the rebels with troops and weapons, which Russia denies.
Ukraine is also concerned about what actions the new United States President will take in the region.
The European Union and US under Barack Obama imposed sanctions on Russia over the conflict, as well as its annexation of Crimea.
Donald Trump has said he is in the early stages of considering lifting US sanctions on Russia.
Ukrainian President Petro Poroshenko has called on foreign powers to put more pressure on Russia.
For its part, Russia says the violence in Ukraine shows the need for swift cooperation between the U-S and Russia.
The conflict has killed nearly 10,000 people since 2014.
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