A deadline imposed by Kiev for pro-Russian rebels to leave buildings in the eastern Ukrainian city of Donetsk has passed.
Earlier Tuesday, a Ukrainian government spokesman told Fox News that the rebels should flee or face "precision weaponry."
The Ukrainian government said the pro-Russia separatist militants in Donetsk had until 4 p.m. local time to leave the buildings they are occupying. An hour after the deadline passed, there were no signs of military action, according to a Fox News reporter in the city.
A spokesman with Ukraine's "anti-terror operation" said that if the rebels do not leave the buildings, those targets would be "hit with precision weaponry."
Fox News also confirmed from another source that Ukraine may use military action Tuesday against the pro-Russian rebels on the same day Ukrainian President-elect Petro Poroshenko said in an interview that the military operation in Donetsk would be over in "a matter of hours."
Donetsk mayor Oleksandr Lukyanchenko said a total 40 people, including two civilians, were killed Monday after troops repelled a rebel attempt to seize control of the airport, Ukraine's second-largest.
Local morgues were overflowing with bodies and rebel leaders said Tuesday that the death toll could rise up to 100.
The bodies of about 30 insurgents were brought Tuesday to a hospital morgue in Donetsk, said Leonid Baranov of the separatist Donetsk People's Republic, who was at the Kalinin morgue. The fighters had been wounded and were being transported to a hospital in a truck when it was shot up by government forces, Baranov said.
Baranov said up to 100 rebels were probably killed in Monday's fighting, adding that many bodies had not yet been recovered because they were in areas under government control.
By Tuesday morning, Donetsk airport was under full government control, Ukraine's acting Interior Minister Arsen Avakov said, adding that dozens of insurgents may have been killed but government forces did not suffer any casualties.
The Organization for Security and Cooperation in Europe, meanwhile, said it had lost contact with one of its four-man monitoring teams in Donetsk on Monday evening. There was no immediate claim of responsibility but rebel groups have previously kidnapped OSCE monitors in Ukraine.
Earlier Tuesday, a group of unidentified men stormed Donetsk's main ice-hockey arena, which was to host the 2015 world championships, and set it ablaze, according to the mayor's office.
In the neighboring Luhansk region, the Ukrainian Border Guards Service said that its officers engaged in a gunbattle with a group of gunmen who were trying to break through the border from Russia. It said one intruder was wounded and the border guards seized several vehicles loaded with Kalashnikov assault rifles, rocket grenade launchers and explosives.
Speaking at a televised government session on Tuesday, Vitaly Yarema, a deputy prime minister in the interim cabinet, said the "anti-terrorist operation" in eastern Ukraine will go on "until all the militants are annihilated."
The battles in Donetsk came just as billionaire candy magnate Petro Poroshenko claimed victory in Sunday's presidential vote. Poroshenko has vowed to negotiate a peaceful end to an insurgency in the east, where rebels have seized government offices and fought Ukrainian troops for more than a month.
Poroshenko described the separatists as "Somali pirates," saying that arms should be used against "killers and terrorists," but he also indicated that he wants a quick end to the military operation in the east.
"The anti-terrorist operation cannot and should not last two or three months," he said. "It should and will last hours."
Poroshenko, known for his pragmatism, supports building strong ties with Europe but also has stressed the importance of mending relations with Moscow. Upon claiming victory, he said his first step as president would be to visit the east.
He said he hoped Russia would support his efforts to bring stability and that he wanted to hold talks with Moscow.
Russia welcomed his intention to engage in talks with people in the east and said it would be ready to work with Poroshenko.
But in Moscow, Russian Foreign Minister Sergey Lavrov voiced strong concern Tuesday about the decision to intensify the military operation in the east and called for an immediate end to fighting.
Lavrov warned Poroshenko against trying to win a quick military victory before his inauguration, saying that it would be "unlikely to create favorable conditions for a hospitable welcome in the Donetsk region." He promised that Russia will be Poroshenko's "serious and reliable partner" if he moved to negotiate an end to hostilities.
Poroshenko is yet to be sworn in and the date for his inauguration hasn't yet been set. The interim government, meanwhile, pledged to press ahead with the operation against insurgents, which has angered local residents, many of whom see the authorities in Kiev as nationalists bent on repressing Russian speakers in the east.
Russia has denied accusations by the Ukrainian interim government and the West that it has fomented the insurgency in the east. Russian President Vladimir Putin has stonewalled the insurgents' appeal to join Russia and welcomed the Ukrainian presidential election in an apparent bid to de-escalate tensions with the West, which has plunged to a post-Cold War low after Russia's annexation of Crimea.
But Russia has kept pushing for Ukraine to decentralize its government, which would give more power to regions, including those in the east, and wants Kiev to withdraw its troops from the area.
Fox News' Greg Palkot and The Associated Press contributed to this report.