Ukrainian forces achieved some success around Bakhmut but suffered "significant" losses in personnel and heavy equipment, CNN wrote on June 8, citing U.S. officials.
Although Kyiv's troops managed to overrun Russians in some areas, they encountered heavier resistance than expected, according to the report.
Russian troops prepared strong defensive lines, armed with anti-tank missiles, mortars, and grenades, and covered by mines in some areas.
Ukraine's losses included a number of U.S.-supplied MRAP armored personnel vehicles, which are designed to withstand mines and improvised explosive devices (IED).
Kyiv hasn't commented on the report.
Deputy Defense Minister Hanna Maliar reported Ukrainian offensive operations near Bakhmut and "other places" on June 5. A day later, she announced advances of 100-1,600 meters in various parts of the eastern front.
On June 8, Maliar said that Russian forces are attempting to go on the offensive around Bakhmut and other areas in Donetsk Oblast but without success.
The battles in the country's east happen against the backdrop of media reports that Kyiv launched its long-awaited counteroffensive in southeastern Zaporizhzhia Oblast.
Russia claimed that the counteroffensive began already on June 5, adding that its forces had successfully repelled it without providing evidence, while Ukraine's officials refuted their claim.