Russia said on Wednesday that some parts of a possible peace deal with Ukraine were close to being agreed after Kyiv agreed to discuss neutrality, raising hopes of an end to the biggest war in Europe since World War Two.

“Neutral status is now being seriously discussed along, of course, with security guarantees,” Foreign Minister Sergei Lavrov said told RBC news.

“Now this very thing is being discussed in negotiations — there are absolutely specific formulations which in my view are close to agreement,” Lavrov said.

He said that President Vladimir Putin had spoken about neutrality, along with security guarantees for Ukraine without Nato enlargement, as one possible variant in February.

Lavrov cautioned that the negotiations were not easy but that there was “some hope of reaching a compromise”.

Ukraine has also made cautious positive statements on peace talks. It says it is willing to negotiate to end the war, but will not surrender or accept Russian ultimatums.

Lavrov said key issues included the security of people in eastern Ukraine, the demilitarisation of Ukraine and the rights of Russian-speaking people in Ukraine.

Ukraine launches counteroffensives

Meanwhile, Ukrainian presidential adviser Mykhailo Podolyak wrote on Twitter on Wednesday that the country's armed forces are launching counteroffensives against Russian forces “in several operational areas”.

“This radically changes the parties' dispositions,” he added, without giving details.

Reuters was unable immediately to verify his comments.

In an update on the war, the general staff of Ukraine's armed forces referred to the “high intensity of hostilities” but did not say where fighting was heaviest.

Ukrainian officials also made clear that the death toll was rising from the war that began when Russia invaded Ukraine on Feb 24.

The emergency service in Ukraine's eastern region of Kharkiv region said on Wednesday that at least 500 residents of the city of Kharkiv have been killed.

Prosecutor General said on Iryna Venediktova said on Facebook that 103 children have been killed so far in the war.

Russian forces have struck more than 400 educational establishments and 59 of them have been destroyed, she said.

The governor of the Chernihiv region in northern Ukraine said there was no electricity in the region's main city, Chernihiv, or in some other settlements in the area.

But Governor Viacheslav Chaus said Ukraine's armed forces “are powerful and inflict powerful blows on the Russian enemy every hour”.

20,000 escape Mariupol

Meanwhile, About 20,000 people have managed to escape the besieged port of Mariupol in private cars, the Ukrainian interior ministry said, but hundreds of thousands remain trapped by Russian shelling, many without heating, power or running water.

Russian forces have focused on Mariupol, constantly bombarding it for the past two weeks, as it is a key city on the Azov Sea coast which they must gain control of to push further west.

Ukrainian officials estimate that more than 2,500 residents have been killed in the fighting and at least 200,000 are in urgent need of evacuation.

Read: Five scenarios for Ukraine after invasion

Moscow yet to capture Ukraine's 10 big cities

Moscow has not captured any of Ukraine's 10 biggest cities following its invasion that began on Feb 24, the largest assault on a European state since 1945.

Russian forces are struggling to overcome the challenges posed by Ukraine's terrain, according to Britain's Ministry of Defence in an intelligence report.

Russian troops have remained largely tied to Ukraine's road network and have demonstrated a reluctance to conduct off-road manoeuvres. The destruction of bridges by Ukrainian forces has also played a key role in stalling Russia's advance, it said.

And Russia's continued failure to gain control of the air has drastically limited its options.

“The tactics of the Ukrainian Armed Forces have adeptly exploited Russia's lack of manoeuvre, frustrating the Russian advance and inflicting heavy losses on the invading forces,” said the report.

Russia calls its actions a “special military operation” to demilitarise and “denazify” Ukraine. Ukraine and Western allies call this a baseless pretext for a war that has raised fears of wider conflict in Europe.

US President Joe Biden will make his first visit to Europe since Russia invaded Ukraine to discuss the crisis with Nato allies next week, the White House said.

Just over three million people have now fled Ukraine, according to the United Nations, with over 1.8m arriving in neighbouring Poland. Its prime minister and those of Slovenia and the Czech Republic were in Kyiv on Tuesday to show solidarity.

In Kyiv, around half of the 3.4m residents have fled and some spend nights sheltering in metro stations.