Pro-Russian rebels have tried several times in recent weeks to take the airport, which lies to the north-west of the city.
The insurgents continued an offensive – begun on Wednesday, October 1 – on “a broad front”, the Ukrainian military says.
An official truce in place in eastern Ukraine has often been violated.
The cease-fire was called on September 5, but, on October 1, four people were killed by a shell which landed on a school in Donetsk, and six died when a minibus was hit.
A spokesman for what the Ukrainian government calls its anti-terrorist operation said Ukrainian forces repelled four attacks on the airport on Wednesday evening.
A T-64 tank was destroyed and seven rebels were killed, Vladyslav Seleznyov told Kanal 5 TV.
The rebels used tanks, multiple-launch rocket systems, artillery and mortars, Vladyslav Seleznyov said, resuming their attacks on Thursday morning with small-arms fire.
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