A spokesman later said the border service had "preliminary information" that 15 separatists had been killed in the fighting.
The insurgency in Ukraine has escalated in the past two weeks, killing scores and prompting some families to flee their homes.
Ukrainian forces have intensified their campaign against separatists, who hold key administration buildings in several towns in eastern Ukraine including Donetsk, since the landslide election of president-elect Petro Poroshenko on May 25.
Poroshenko was in France on Friday to attend the commemoration of the Allied D-Day landings in World War Two. Informal talks on Ukraine's crisis may take place among several world leaders who are attending, including US President Barack Obama.
Poroshenko, who will be sworn into office on Saturday, has said he will soon announce a peace plan for the region, where pro-Russian rebellions broke out after the ouster of Moscow-leaning President Viktor Yanukovich and the installment of a new pro-European government.