Fighting for the airport has raged for months as the insurgents have tried to dislodge the government forces using it to shell rebel positions in Donetsk, the largest rebel-held city.
At least nine people were killed in the crossfire Wednesday in residential areas near the airport.
Civilian and military casualties have continued to rise in eastern Ukraine despite a cease-fire Sept. 5 and a second agreement Sept. 20 that spelled out how to create a buffer zone.
While that helped to enforce the cease-fire in areas where Ukrainians troops and rebels chose to retreat, non-stop fighting has continued at the airport and other strategic locations.
While the Donetsk airport building has been gutted by shelling, its long runway would let the rebels handle heavy cargo planes carrying supplies, instead of relying solely on truck convoys from Russia.
While it was impossible to get close to the airport Wednesday because of the fighting, an AP reporter in Donetsk saw that artillery fire hitting the airport was coming from government-held positions outside the city — an indication that Kiev may no longer hold the airport.
But Ukraine's National Security and Defense Council spokesman, Col. Andriy Lysenko, told journalists in Kiev the airport was still under the control of government troops who were "brilliantly carrying out their duty."
Rebel leader Alexander Zakharchenko, however, was quoted by the Interfax news agency as saying the rebels now control 90 percent of the airport.
"In two — or maximum three — days, the Donetsk airport will come under our control," he said.
The increasingly violent battle has pounded Donetsk's northern neighborhoods in a conflict that has claimed at least 3,500 lives since April, a month after Russia annexed Crimea.
One shell exploded Wednesday in a school courtyard in northern Donetsk and an AP reporter saw bodies of three people killed by it.
Shortly afterward, another shell hit a nearby bus stop, killing three people seen by the AP.
A minibus that was also hit was still burning hours later.
The Donetsk city council said three people were killed at the school — a teacher and two parents — and 70 schoolchildren were there at the time.
It also said six people were killed and several wounded at the bus stop.
The school's shelling drew a strong condemnation from the United Nations.
Stephane Dujarric, spokesman for the U.N. secretary-general, said that "targeting of schools is unacceptable in any circumstances."
"All parties should redouble their efforts to find a diplomatic solution," he added.
In Moscow, Russian Foreign Minister Sergey Lavrov urged the West to look into the allegations of mass graves of civilians in eastern Ukraine.
"It's a terrible tragedy. It's an obvious war crime," he said at a briefing.
"We hope that Western capitals will not keep silent about these outrageous facts."
Lavrov said more than 400 bodies have been found in mass graves near Donetsk, but Andrei Purgin, a rebel leader there, clarified Wednesday that the figure referred to the total number of unidentified bodies in Donetsk morgues.
Purgin said the bodies of nine civilians have been found in a mass grave near Donetsk.
Authorities in Kiev have rejected rebel allegations that those people were killed by government forces.
In Washington, State Department Spokeswoman Jen Psaki said the U.S. was aware the of reports.
"We've certainly seen the disturbing reports of this discovery of additional graves in the Donetsk region. We call for Russia-supported separatists to allow access to the site and we would support a full and thorough investigation," Psaki told reporters.
Source: The Huffington Post