Footage posted on social media shows two Soviet-era planes moving along the runway before one shoots forward and rams into the tail of the other.
A military spokesman said two military transport planes collided while landing at Khartoum airport after training flights on Wednesday morning.
The aircraft, one an Antonov 26 and another an Antonov 32, suffered severe damage but there were no casualties, he said.
However, the Al-Sudan al-Youm news website issued a conflicting report, saying eight people were injured in the crash.
International flights were being diverted to Port Sudan airport, a civil aviation department spokesman said. The airport has since reopened.
Abdel-Hafez Abdel-Rahim said the closure of the airport was precautionary in line with international regulations.
Additional reporting by agencies
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