violence yet in the flashpoint city.
An additional 31 people have been injured, including four civilians, according to the website of the Donetsk mayor, Alexander Lukyanchenko. Two of the fatalities are civilians.
The conflict at Donetsk International Airport broke out only hours after newly elected Ukrainian President Petro Poroshenko said he'd potentially like to negotiate a way out of the crisis.
After Ukrainian forces moved in against the militants Monday, the deadly assault continued overnight.
The airport remained closed Tuesday despite an easing in the gunfire, as conflicting accounts emerged of how many had lost their lives.
The Donetsk mayor's website didn't specify how many of the 40 killed in the airport standoff were separatists.
But a spokeswoman for the separatist self-declared "Donetsk People's Republic" (DNR) told CNN that 35 separatists had been killed and about 60 injured in Monday's fighting.
Although the separatists earlier claimed they controlled the airport, it became clear as Tuesday wore on that the Ukrainian military had taken charge.
The official website of Ukrainian Interior Minister Arsen Avakov said late Tuesday the "Airport in Donetsk is fully under our control."
The occasional exchange of gunfire and blasts could be heard from the airport but it was not clear if the shots were fired by military as they maintained a perimeter or whether separatist forces were still present within its territory.
Two blown-out trucks nearby appeared to have been hit by heavy weapons. Human remains were still visible, suggesting this may have been the cause of some of the casualties Monday.
The separatist movement in Donetsk believed it was offered a three-hour 'truce' Tuesday to leave the city of Donetsk, according to a spokeswoman for the DNR who asked not to be identified to avoid possible arrest. The truce was offered between 1 and 4 p.m. local time (6 a.m. and 9 a.m. ET), she said.
The DPR learned of this truce online, the spokeswoman told CNN, adding that Ukrainian armed forces were threatening to bomb separatist strongholds in the city if they failed to leave.
The Ukrainian government denied offering rebels any such truce. The Ukraine Government's Anti-Terror Operation (ATO) told CNN there is a longstanding offer of amnesty to any separatist who turns himself in and gives up his weapons, unless he (or she) is guilty of murder.
Morgue piled with bodies
A CNN team at a morgue in Donetsk saw a large pile of separatist militant bodies, many of which had been torn apart by shrapnel and explosions.
Doctors there said 31 bodies had been brought in with different types of injuries, from bullet wounds to those caused by heavy weapons and explosions. The remains included the body of a woman civilian.
Doctors also said some locals had arrived during the morning to identify and collect their relatives from among the dead.
The airport clashes marked the worst violence that this key population center in eastern Ukraine has seen since the start of the crisis. A statement posted on the mayor's website Monday advised residents to stay in their homes as sounds of gunfire and explosions cracked through the air.
Elsewhere in the Donetsk region, the Organization for Security and Cooperation in Europe said it has had no contact with one of its Donetsk-based teams since around 6 p.m. (11 a.m. ET) Monday .
The team of four -- of Swiss, Turkish, Estonian and Danish nationality -- was on a routine patrol east of Donetsk when it was last heard from. The OSCE says it is using contacts on the ground to try to determine where the monitors are.
The last time an OSCE team went missing in Donetsk, its members turned up in the hands of militant separatists in the flashpoint town of Slovyansk. They were freed just over a week later.
In another development, NATO has observed Russian troop movement near the Russia-Ukraine border recently, a NATO officer told CNN. Speaking on condition of anonymity, the officer said this included signs of Russian equipment and supplies being packed or prepared for movement, and that the activity could signal a slow or staged withdrawal of forces.