One difficulty in determining the exact start of a counteroffensive, beyond Ukraine’s operational security measures, is that the fighting could well begin with feints or diversions that are hard to decipher.

The Russian Ministry of Defense announced on Monday that a major Ukrainian operation had begun at five locations along the front in one eastern region, Donetsk, and said it had repelled the assaults and inflicted casualties on the Ukrainian forces. Moscow’s reports could not be corroborated.

Russian bloggers affiliated with the Russian military went further, saying Ukraine had advanced in some areas and that an even stronger attack had begun early Monday morning near the town of Velyka Novosilka.

There was no immediate response from the Ukrainian authorities to the Russian claims.

Buttressing the reports that the counteroffensive may have started, American military analysts said they believed Ukrainian units had begun an initial thrust to determine the position and strength of Russia’s forces. Such moves would be a traditional tactic of the kind Americans have been training Ukrainian forces to undertake.

Much is riding on Ukraine’s counteroffensive to reclaim territory lost to Russia and free Ukrainians from living under occupation. Success could help support Kyiv’s push for longer term commitments for additional military aid and security guarantees from the West. It also could strengthen the hand of President Volodymyr Zelensky in any peace talks with Russia.

Failure, or a lack of dramatic quick progress, could complicate Ukraine’s ability to get further assistance from the West and make Kyiv’s push for additional security guarantees at the NATO summit this summer more difficult.

Whatever precisely was happening, the relative calm that most of the long battle lines stretching across Ukraine had been experiencing was broken.

The front in southern and eastern Ukraine has been largely static for months, with the exception of intense fighting in Bakhmut and some testing operations by small Ukrainian units. In northeastern Ukraine, pro-Ukrainian forces also began cross-border raids into Russia last month.

The initial reports of the fighting Sunday night and Monday morning were of movements of larger units, a potential signal that Ukraine had pushed its NATO-trained forces into the fight.

Two unverified Russian reports said Ukrainian forces had pushed through Russian defenses in two areas, in the Donetsk and Zaporizhzhia regions.

By early Monday, a Russian Telegram channel that reports on the war said Ukrainian forces had breached a first line of Russian defenses near Velyka Novosilka in Donetsk and reclaimed two villages as Russian forces fell back to reserve positions. The fighting at another location was confirmed by the head of the regional occupation authority in the Zaporizhzhia region, Vladimir Rogov, in a post on Telegram.

Ukrainian officials have not told their American counterparts exactly when the counteroffensive will start, but have provided them with a time frame during which they intended to begin their push against Russian forces. Sunday was within that time frame, said U.S. officials, who spoke on the condition of anonymity to discuss sensitive intelligence.

The renewed fighting, and possible start to the counteroffensive, came just days before the D-Day anniversary. It was largely determined by the speed of training and equipping new units, rather than any anniversary. But the symbolism of timing a counteroffensive in Ukraine to the operation that began the Western allies’ drive to retake Europe would hardly be lost on Mr. Zelensky, a master communicator who has won over many countries to his cause.

The Pentagon has trained new Ukrainian units with the hope that they have the power to turn the tide of the battle, but some American officials have noted that dug in Russian defenders could prove difficult to dislodge.

Since March, smaller Ukrainian units of tanks and armored personnel carriers had conducted operations along the battle lines in Zaporizhzhia province. The true beginning of any offensive push, if that is what began Sunday, would involve larger units not just conducting probing attacks but actively trying to break through Russian defenses.