Ukrainian government troops have advanced from the north and south in a bid to cut off the fighters and weapons that Kiev says are flowing over its border from Russia.
So far, though, Moscow has given little sign it intends to respond to repeated requests from separatists and send in troops to assert control in parts of Ukraine's east, as it did with Crimea in March.
The Kremlin appears to be keeping its support at a level that won't provoke harsher economic sanctions from the West but will keep the conflict bubbling to use as leverage over Kiev by sending more fighters and weapons, some Western diplomats and analysts say.
While the West has warned Russia it risks more penalties if it doesn't help stabilize Ukraine's east, some European countries have voiced reluctance to step up sanctions, given deep trade ties with Moscow.
"Russia's intention is instability, not occupation or even separation," said Adrian Karatnycky, senior fellow at the Atlantic Council, a Washington-based think tank.
"So a low-intensity guerrilla war and disruptive occupation of cities is what they aimed for and what they are employing."
Ukrainian and U.S. officials say Russia in recent weeks has allowed fighters, tanks and other heavy weaponry across the border, which Russia has denied.
In one of the deadliest clashes to date, separatists last week downed a Ukrainian military transport plane with heavy machine-gun fire and shoulder-launched missiles, killing all 49 people on board.
But separatist leaders said Tuesday that their manpower and weapons weren't enough to hold off a Ukrainian military operation in recent days aimed at sealing the border.
After regaining control of the large port town of Mariupol on Friday, Ukrainian forces on Tuesday advanced toward the outskirts of Luhansk, the capital of Ukraine's easternmost region and a separatist stronghold.
Ukrainian President Petro Poroshenko said Monday that a cease-fire could be declared as early as this week if government forces retake control of the border.
Ukraine would then disarm the rebel fighters and grant amnesty to anyone not guilty of serious crimes, he said.
"They are methodically cutting off the rebel regions from the border with Russia," said a rebel commander known as Igor Strelkov in a video posted by Russian newspaper Komsomolskaya Pravda.
Mr. Strelkov said Ukrainian forces were advancing from the north and south along the border with Russia at a speed of up to nine miles a day.
A Ukrainian army spokesman said its forces had killed or wounded around 30 rebel gunmen in the Luhansk region Monday.
Mr. Strelkov complained that Russia had missed its chance to send in "peacekeepers" after separatists declared victory in a May 11 secession referendum in parts of Ukraine's east.
Kiev declared the vote illegal and riddled with irregularities.
"We can only defend," he said, as explosions boomed in the background.
"Unfortunately, our opponent is superior to us on all parts of the front."
In Moscow for meetings with Russian officials, another separatist leader thanked Russia for sending humanitarian aid but said that rebels needed military aid and advice to form a functioning government.
"We are not professional politicians or revolutionaries," Denis Pushilin, a leader of the self-proclaimed Donetsk People's Republic, told a news conference in the Russian capital Tuesday.
"We are looking to the Russian Federation for help."
Mr. Pushilin said that there was a "flow of volunteer [fighters] from Russia and European countries" but that they remained outnumbered and outgunned by the Ukrainian army.
Mr. Pushilin said he had met with top Russian officials during his visit to Moscow, including presidential advisers Vladislav Surkov and Sergei Glazyev.
The officials' offices didn't respond to a request for comment, and the meetings weren't widely reported in state-controlled media.
Russian state television did show Mr. Pushilin's meeting last week with radical nationalist lawmaker Vladimir Zhirinovsky over plates of biscuits.
The Kremlin's perceived inaction has drawn criticism among some radical nationalists in Moscow.
"Putin is dragging his feet," Alexander Dugin, a pro-Kremlin ideologue who has been one of the closest allies of the separatists in Moscow, wrote in a Facebook post Monday.
Ukrainian officials said Russia has started moving some military units toward the border after they had been ordered back to their bases last month.
A spokesman for Ukraine's foreign ministry said Russia was also forming groups of volunteer fighters in a border region to be sent into Ukraine.
The spokesman, Yevhen Perebyinis, said Russia's continued interference means the West should implement tougher economic sanctions.
"We're not expecting anything [positive] from the Russian side. They have made absolutely clear that they don't intend to stop their activities, and the only thing that can stop the Russian side is tougher sanctions," he said.
Amid the continuing violence, German Foreign Minister Frank-Walter Steinmeier renewed threats of more Western sanctions.
He said at meeting with his Baltic counterparts in Estonia on Tuesday that European Union leaders may consider additional sanctions against Russia when they meet in Brussels late next week.
"If Russia is responsible for an extended destabilization of Ukraine, further decisions will be unavoidable," Mr. Steinmeier said in comments reported by Reuters and confirmed by the Foreign Ministry in Berlin.
"We haven't conclusively discussed whether or not we are in that situation currently."
European leaders have said for months that they are prepared to enact sanctions against sectors of the Russian economy.
Those threats quieted in recent weeks as diplomats tried to convince the Kremlin to work more closely with the West in resolving the crisis.
But on Tuesday, Mr. Steinmeier sounded a more pessimistic note than days ago, when he voiced hope that a change in tone from Moscow augured progress.
He said the recent failure of natural-gas negotiations between Moscow and Kiev and the deadly downing of the Ukrainian military plane represented significant setbacks.
"For all the visible effort and small steps forward, the last weekend showed that we are far from a political solution to Ukraine crisis," Mr. Steinmeier said, according to Reuters.
Source: The Wall Street Journal