If evacuation takes place, officials say up to 8,000 people could be removed each day from the government-held town, which has no water or electricity.
Shelling and the deaths of several more people were reported by both sides on Tuesday.
Each blames the other for the upsurge in violence.
It erupted despite an attempt to renew a ceasefire last month.
Ukrainian forces say the outbreak began when rebels launched an attack on Avdiivka, which borders land controlled by the separatists.
Seven soldiers and a number of rebels have been killed in recent days, and there have been civilian casualties on both sides but precise numbers are unclear.
For several hours 200 coal miners were trapped underground on Tuesday when shelling cut power at a pit in the rebel-run area of Donetsk.
Engineers at Zasyadko eventually managed to get generators working and started bringing the miners back to the surface.
"Everything is going bad," a resident in Avdiivka told the BBC. "People are scared and try to go out. It's very cold. There are bread queues. Only a few shops are open," said the woman, who identified herself as Nadiya.
On Tuesday, the head of the Kiev-appointed administration, Pavlo Zhebrivsky, said plans were being made for an evacuation of residents.
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"As of now, we can evacuate up to 8,000 people in the course of the day. The region's towns are ready to receive up to 9,000 people," he was quoted as saying by Interfax news agency.
The population of the town is uncertain but is believed to range between 16,000 and 22,000.
Officials say the evacuation will go ahead if the fighting escalates further. About 10 tonnes of food will soon arrive in the city, they add.
Recaptured from pro-Russian rebels in 2014, Avdiivka is a key stronghold for the Ukrainian army which it is desperately trying to retain.
The town is close to rebel-controlled Donetsk and important roads and intersections used by the rebels to transport machinery and ammunition.
Just as important is the town's coking and chemical plant - the biggest of its kind in Europe. If it fell into rebel hands it would cut much-needed supplies to Ukraine's steel industry.
The renewed violence coincided with President Donald Trump's first phone-call with Russian President Vladimir Putin since he took office.
The government in Kiev fears it may lose US support during Mr Trump's presidency. According to the Kremlin, Mr Trump and the Russian leader agreed to a "partner-like co-operation" on issues including Ukraine.
While Ukraine accuses rebels of starting the violence, Russian government spokesman Dmitry Peskov on Tuesday blamed government forces.
"Such aggressive actions, supported by the armed forces of Ukraine, undermine the aims and the task of realising the Minsk accords," he said on Tuesday.
A ceasefire was agreed in Minsk in February 2015 but there have been frequent violations. The latest truce began on 23 December.
More than 9,700 people have died since the conflict erupted in 2014, as Russia annexed Ukraine's southern Crimea peninsula, and pro-Russian rebels later launched an insurgency in the east.
The US and EU imposed sanctions on Russia in response to its actions in eastern Ukraine. Russia has denied backing the rebels.
Ukrainian officials are preparing for a possible evacuation of the eastern frontline town of Avdiivka amid renewed fighting with pro-Russian rebels.