The West has also accused Russia of most likely providing the insurgents with surface-to-air missiles that may have been used to shoot down a Malaysia Airlines passenger jet over rebel-held territory on July 17, killing all 298 people on board.
A Dutch plane carrying victims' remains from the disaster took off from Kharkiv on Monday. The plane was also carrying personal belongings of victims back to the Netherlands.
The Interfax news agency reported Monday that Russia's air force began military drills in central and western regions of the country, a move that could spark further fears that Moscow is ready to flex its military muscle in Ukraine.
The drills will start Monday and last through Friday, air force chief Igor Klimov was reported as saying, and will involve more than 100 fighter jets and helicopters.
Vasily Malayev, head of the Federal Security Service's border patrol in the Rostov region, said that 438 Ukrainian soldiers had crossed into Russian territory on Monday. He said the Russian side had allowed the soldiers to safely enter the country overnight.
The ITAR-Tass news agency quoted Malayev as saying that 180 of the soldiers had requested to be sent back to Ukraine, and that the Russian side would help them travel back.
Russia's Defense Ministry couldn't immediately be reached for comment.
The Ukrainian military confirmed that part of a brigade had most likely crossed into Russian territory, although it disputed Russia's version of events and wouldn't say how many soldiers went over.
A spokesman for the Ukrainian military operation in the east, Oleksiy Dmitrashkovsky, said troops from the army's 72nd Brigade were penned into their position and came under a sustained barrage of fire from separatist forces. Rebel fighters used tanks, mortars, artillery and Grad missile launchers over four hours, Dmitrashkovsky said, and eventually the brigade was forced to divide up into two sections.
"One was meant to break out and join forces with a support unit. The other unit had the task of providing fire cover," Dmitrashkovsky said. "In doing that they fired their weapons until no ammunition remained, after which they abandoned their position and reached a place near a border crossing on Russian territory."
Dmitrashkovsky said it was too early to confirm how many soldiers had crossed into Russia.
"We do not have such information. The Russians are capable of claiming anything they want," he said.
Yuras Karmanau contributed to this report from Donetsk, Ukraine.