Government forces lost more ground to Russian-backed separatists in heavy fighting in eastern Ukraine on Sunday, a day after European leaders threatened to impose more sanctions on Moscow if it doesn’t end its support for the rebels.

Ukrainian President Petro Poroshenko said he hoped a new round of talks with rebels and Russian officials in Minsk on Monday could yield progress toward a cease-fire. It wasn’t clear whether Kiev was ready to accept a deal that would leave the separatists in control of their strongholds in the cities of Luhansk and Donetsk.

Previously, Poroshenko had vowed to defeat them unless they surrendered. But after dramatic reversals on the battlefield in recent days — following what Kiev and the North Atlantic Treaty Organization say was an incursion of more than 1,000 Russian troops to fight alongside the separatists — his negotiating position seems weakened. European leaders were cool to his appeals for military aid and held off immediately imposing more sanctions to give Russia a chance to change course.

The Kremlin on Sunday stepped up its calls for Kiev to lay down its arms and begin talks, effectively recognizing the Moscow-backed rebels’ hold over a major industrial region. Moscow also hinted at support for the rebels’ demands for independence.

President Vladimir Putin, in an interview on state television Sunday, said Kiev should begin “substantive, meaningful negotiations, and not on technical issues, but on the political organization of society and statehood in the southeast of Ukraine.”

A Kremlin spokesman later told reporters that the reference to “statehood” wasn’t meant as an endorsement of the separatists’ demands for independence from Ukraine. But the spokesman said Ukraine would have to discuss autonomy with a region he referred to as “Novorossiya.” That tsarist-era word, referring to a swath of territory that includes much of present-day southern and eastern Ukraine, has become a rallying cry for Russian nationalists. It had faded from official Kremlin statements since the spring but reappeared this week.

The latest events on the ground were a dramatic reversal for Kiev, whose forces had been close to retaking the rebel strongholds in Donetsk and Luhansk in early August, officials said. The introduction of several units of Russian troops over the past week or so has dramatically turned the tide.

Security officials in Kiev say the combination of Russian troops, tanks and antiaircraft systems — which helped neutralized Kiev’s advantage in air power — allowed the separatists to reopen key supply lines to Luhansk and Donetsk and open a new front for the strategic port city of Mariupol. Moscow denies deploying troops, though rebels say Russian military personnel are fighting on their side, though while on vacation.

On Sunday, Ukraine exchanged 10 Russian paratroopers captured on its territory early last week for 63 Ukrainian troops who had crossed into Russia after becoming surrounded. Moscow has said the paratroops crossed into Ukraine by accident.

Ukraine’s setbacks have prompted alarm among Western officials, but little signs of outright military aid.

Poroshenko returned Sunday from a European Union summit in Brussels after the EU agreed to draw up options within a week for possible new sanctions against Russia. He left, though, without any promise of military support from the bloc’s 28 governments.

An expanded version of this report appears on WSJ.com.