More than half the Ukrainian fighters in the bunkers below the Azovstal steel plant have surrendered, the leader of the Russian-backed rebel Donetsk region said on Thursday, and the International Committee of the Red Cross is registering them.

The most devastating siege in Russia's invasion is drawing to a close after nearly 1,000 Ukrainian fighters laid down their arms this week, giving themselves up to pro-Russian forces.

The ICRC said it had registered those surrendering from the plant so that it can track them and ensure their families can remain in contact with them after their capture.

The wounded have been given medical treatment while those who are fit have been taken to a penal colony and are being treated well, said Denis Pushilin, head of the breakaway Donetsk People's Republic.

"More than a half have already left - more than half have laid down their arms," Pushilin told the Solovyov Live internet television channel.

"Let them surrender, let them live, let them honestly face the charges for all their crimes," Pushilin said.

Ukrainian officials have declined to comment publicly on the fate of the fighters, saying it could endanger rescue efforts.

Ukrainian President Volodymyr Zelenskiy casts the fighters - which include Ukrainian army soldiers and members of the Azov Regiment - as heroic defenders of the motherland against foreign occupiers.

Russia, though, says the Azov Regiment, which began as an extreme-right nationalist paramilitary organisation, is a group of radically anti-Russian nationalist fighters and casts them as modern-day Nazi sympathisers.

The regiment, formed in 2014 as a militia to fight Russian-backed separatists, denies being fascist, racist or neo-Nazi, and Ukraine says it has been reformed away from its radical nationalist origins to be integrated into the National Guard.

Asked about reports that Captain Sviatoslav Palamar, 39, a deputy commander of the Azov Regiment, had surrendered, Pushilin declined comment. Palamar told Reuters last month that his forces would fight for as long as needed.

The ultimate fate of the fighters is unclear.

The Kremlin said the combatants would be treated in line with international norms, though some Russian lawmakers demanded they be put on trial and one demanded they face the death penalty.

The ICRC said it had registered those leaving the Azovstal plans to track those captured and keep in touch with their families.

"The operation continued Wednesday and was still ongoing Thursday," it said. "The ICRC is not transporting POWs to the places where they are held."

Russia's interior ministry put two Azov commanders, Serhiy Velychko and Konstantin Nemychev, on a wanted list.