"Bakhmut would have been taken before the New Year, if not for our monstrous military bureaucracy ... and the spokes that are put in the wheels daily," Wagner head Yevgeny Prigozhin said in one of several messages posted online overnight.
"I think it's (going to be in) March or in April," he said.
To take Bakhmut you have to cut all supply routes. It's a significant task," Prigozhin asserted, adding: "Progress is not going as fast as we would like."
Russia has been trying to encircle the battered industrial city and wrest it ahead of February 24, the first anniversary of what it terms its "special military operation" in Ukraine.
The fierce fighting for the eastern industrial city is now the longest running battle of Russia's intervention and Moscow's key military objective.
The capture of Bakhmut would be a major win for Moscow but analysts say its capture would be mainly symbolic as the salt-mining town holds little strategic value.
Ukrainian forces are determined not to cede any ground ahead of an anticipated counter-offensive in the spring.
Prigohzin, who is close to Putin, said the speed of Russian progress in the grinding battle would depend on whether Ukraine continued to send reserves to hold the city.
His private fighting force, which has recruited prisoners from across Russia with the promise of amnesty, has claimed a lead role in recent battles in east Ukraine.
He announced last week that Wagner would no longer be tapping prisons to fill its ranks and on Thursday warned this would also impact the fighting.
"Of course, at some point the number of units will drop and as a result the number of tasks that we can perform will not be what we want," he added.
Wagner's claims to have captured ground without help from the regular army has spurred friction with senior military leadership.
Prigozhin has previously accused the Russian military of attempting to "steal" victories from Wagner, a sign of his rising clout and the potential for dangerous rifts in Moscow.
