Mutinous Russian mercenary fighters who surged most of the way to Moscow have agreed to turn back to avoid bloodshed, their leader said on Saturday, in a de-escalation of what had become a major challenge to President Vladimir Putin's grip on power.

The fighters of the Wagner private army run by former Putin ally Yevgeny Prigozhin were already most of the way to the capital, having captured the city of Rostov and set off on an 680-mile race to Moscow.

In an audio message, Prigozhin said the fighters would return to base because of the risk of blood being spilled.

A deal to halt further movement of Wagner fighters across Russia in return for guarantees of safety for the rebels was brokered by Belarusian President Alexander Lukashenko, his office said.

Earlier, Prigozhin said his men were on a "march for justice" to remove corrupt and incompetent Russian commanders he blames for botching the war in Ukraine.

In a televised address from the Kremlin, Putin said Russia's very existence was under threat.

"We are fighting for the lives and security of our people, for our sovereignty and independence, for the right to remain Russia, a state with a thousand-year history," he said.

"All those who deliberately stepped on the path of betrayal, who prepared an armed insurrection, who took the path of blackmail and terrorist methods, will suffer inevitable punishment, will answer both to the law and to our people."

Putin later signed a law tightening rules for breaking martial law in places where it has been imposed, the RIA news agency said.

Video obtained by Reuters showed troop carriers and two flatbed trucks each carrying a tank driving 30 miles beyond Voronezh, more than half way to Moscow, where a helicopter fired on them.

Prigozhin, whose private army fought the bloodiest battles in Ukraine even as he feuded for months with the military top brass, said he had captured the headquarters of Russia's Southern Military District in the city of Rostov without firing a shot.

Rostov serves as the main rear logistical hub for Russia's entire invasion force in Ukraine.

Prigozhin said that while his men are just 120 miles from Moscow, he decided to turn them back to avoid "shedding Russian blood."

He did not say whether the Kremlin has responded to his demand to oust Defense Minister Sergei Shoigu. There was no immediate comment from the Kremlin.

Prigozhin has accepted Lukashenko's offer to halt the Wagner group's advance and further steps to deescalate the tensions, Lukashenko's office said, adding the proposed settlement contains security guarantees for Wagner troops. It did not elaborate.

Information from The Associated Press was used in this report.