AVDIIVKA - Clashes between Ukrainian forces and Moscow-backed rebels left two more dead in a flashpoint town Friday and four others elsewhere in bloodshed that has prompted the US to condemn Russia's "aggressive" stance.

The latest reports from both sides put the death toll in fighting since Sunday in the eastern war zone to 32 - a figure that has not been reached since the height of the war in 2014 and 2015.

The US criticism of Moscow will be welcomed in Kiev because it is particularly worried about losing Washington's support as new President Donald Trump seeks to mend ties with Russia's Vladimir Putin.

Ukrainian forces and rebels have been exchanging mortar and rocket fire near the frontline town of Avdiivka just north of the pro-Russian rebels' de facto capital of Donetsk.

The Ukrainian army said the latest fatal victims were a local woman and an emergency worker in Avdiivka. The toll in the industrial town now stands at 27.

Authorities said that three soldiers had also been killed in other parts of eastern Ukraine in which pro-Russian insurgency has raged since April 2014. The rebels also said that shelling by the Ukrainian army had reportedly killed two civilians in Donetsk.

Ukraine and Russia have traded blame over who started the violence and who was responsible for the deaths.

AFP reporters in Avdiivka said Friday the shelling had died down considerably in the town itself and that only sporadic echoes of mortar fire could be heard on its outskirts.

They saw rebels on repeated attack against the Kiev-held blue-collar coke plant town of some 20,000 people throughout the week.

The separatists' sudden assault came after a month-long spell of relative calm in the 33-month war and sharpened the world's focus deadly fighting in the European Union's back yard. The conflict began less than two months after Ukraine ousted its Russian-back leader in April 2014. Moscow responded by annexing Ukraine's Crimea peninsula in March 2014 and then allegedly plotting the eastern insurgency to keep Ukraine under its thumb after its tilt toward the West.

The Kremlin sees most of the former Soviet Union as part of its geopolitical sphere of influence - a stance strongly disputed by the United States and European leaders such as German Chancellor Angela Merkel.

The United States' UN ambassador Nikki Haley on Thursday condemned Russia's "aggressive actions" in Ukraine. Her comments were vital for Kiev because it is concerned that Trump's bid to mend ties with Moscow could have an impact on Washington's stance on the Ukraine conflict.

The shelling in Avdiivka has left nearly 20,000 people without heating or water in freezing weather. The town now has access to water but that pressure systems were too weak to make it accessible in the higher floors of tall buildings.

The United States also said it would not lift sanctions against Russia - as many in Kiev feared it would do under Trump - until the Kremlin returned Crimea to Ukraine.

The Black Sea peninsula of some two million people is home to a tsarist-era Russian naval base and is the bulwark of what Moscow sees as NATO's aggression in eastern Europe.

Ukrainian President Petro Poroshenko has vowed to win back Crimea but has never said how this would be achieved.

The possibility of Russia giving up the region on its own seem extremely remote.

Avdiivka was seized by the insurgents when the conflict began but was recaptured by government forces several months later.

Negotiators from Ukraine and Russia agreed in the Belarussian capital Minsk on Wednesday to call for an immediate truce. The rebels were not present at the meeting but have previously accepted such calls in the past.