Russia’s media carried a statement purported to be by the Wagner head, Yevgeniy Prigozhin, who said Ukrainians were now encircled in the city centre, writes BBC.
Ukraine’s Deputy Defence Minister Hanna Maliar earlier said that “heavy fighting is continuing”.
The claims by both sides have not been independently verified.
In the reported statement late on Tuesday, Prigozhin said: “Wagner units took control of the entire territory of Soledar. A cauldron has been formed in the centre of the city in which urban fighting is going on”.
He stressed that only Wagner fighters - who are not part of the Russian armed forces - were taking part “in the storming” of Soledar.
Meanwhile, Ms Maliar said a few hours before that “the enemy does not pay attention to the large losses of its personnel and continues to actively storm”: “The approaches to our positions are simply strewn with the bodies of dead enemy fighters. Our fighters are defending bravely”.