"An agreement has been reached on the removal of the wounded," the Russian defence ministry said in a statement.
"A humanitarian corridor has been opened through which wounded Ukrainian servicemen are being taken to a medical facility in Novoazovsk."
Ukraine's Deputy Defence Minister Hanna Malyar told Ukrainian television: "Any information can harm the processes that are taking place... Inasmuch as the process is under way, we can't say what's happening right now."
As Russian forces pummelled Mariupol for nearly two months, some civilians and Ukrainian fighters sought refuge in the Azovstal works - a vast Soviet-era plant founded under Josef Stalin and designed with a labyrinth of bunkers and tunnels to withstand attack.
Most civilians were allowed to leave the plant this month after the United Nations and the International Committee of the Red Cross brokered a deal with Russia and Ukraine.
But Ukrainian fighters from the Azov Battalion, a militia set up by Ukrainian ultra-nationalists in 2014 and later incorporated as a regiment in Ukraine's national guard, remain at the plant.
Videos and pictures posted online have shown some with serious injuries.
Relatives appealed on Monday to Turkish President Tayyip Erdogan in Istanbul to help extract the defenders.
Natalia Zaritskaya, wife of a member of the Azov Battalion, told reporters: "The ring around Azovstal has tightened. We can't delay any further. We pin our last hope and believe that through the joint efforts of Turkey, through its President Erdogan, and China, through its President Xi Jinping, and God himself, it is possible to save Azovstal and the people who are there on the cusp of life and death.
"They are in hell. They receive new wounds every day. They are without legs or arms, exhausted, without medicines."
A senior commander of forces holed up beneath the steel mill said he was carrying out a decision by the military high command to save the lives of service personnel but made no mention of surrendering.
"The main thing is to realise all the risks, is there a plan B, are you fully committed to that plan which must allow for fulfilling the assigned tasks and preserve the lives and health of personnel?" Commander Denys Prokopenko said in a video posted on social media.
"This is the highest level of overseeing troops. All the more so when your decision is endorsed by the highest military command."
On Sunday, brightly burning white munitions were seen raining down on the steel works in what a British military expert said looked like an attack with phosphorus or other incendiary weapons.
Meanwhile, Russian President Vladimir Putin appeared to climb down on Monday from Russia's objections to Sweden and Finland joining NATO, saying the Kremlin had no issues with them entering the US-led military alliance they now aim to join in reaction to his invasion of Ukraine.
Putin said Russia would take action if NATO were to move more troops or hardware onto the territory of its new members - steps Finland and Sweden have both already ruled out - but that NATO's expansion itself was not a threat.
"As far as expansion goes, including new members Finland and Sweden, Russia has no problems with these states - none. And so in this sense there is no immediate threat to Russia from an expansion to include these countries," Putin said.
"The expansion of military infrastructure into this territory would certainly provoke our response. What that (response) will be - we will see what threats are created for us," Putin said.
"Problems are being created for no reason at all. We shall react accordingly."
Both non-aligned throughout the Cold War, Finland and Sweden say they now want the protection offered by NATO's treaty, under which an attack on any member is an attack on all.
"We are leaving one era behind us and entering a new one," Swedish Prime Minister Magdalena Andersson said, announcing plans to formally abandon the militarily non-aligned status that has been a cornerstone of Swedish identity for more than 200 years.
"NATO will strengthen Sweden, Sweden will strengthen NATO," she said.
Swedish and Finnish officials have said Putin has only himself to blame for their decisions to join NATO.
The Ukraine invasion, now nearly three months old, has so far been a military disaster for Russia, with its troops forced out of the north and the environs of Kyiv in late March.