Mr Pavkovic (right) was once a Milosevic loyalist
Belgrade's B-92 radio says Mr Kostunica criticised General Nebojsa Pavkovic because of his purported ignorance of a military intelligence operation that led to the arrest of a deputy prime minister and a US diplomat two weeks ago.
Mr Djindjic has denounced the arrest of the deputy prime minister, Momcilo Perisic, as an attempt to discredit his government.
General Pavkovic had sought to distance himself from the arrests, saying he had only been informed a day later - indicating that military intelligence was acting on its own.
Correspondents say Mr Kostunica's demand for the chief of staff to be sacked came as a surprise.
He had earlier stood by the general, a close aide of the former president Slobodan Milosevic, despite pressure from the West to remove him.
Mr Perisic, who was himself army chief of staff during much of Slobodan Milosevic's rule, has denied accusations that he handed classified information to the US diplomat, named in Belgrade as John David Neighbor.
Although Mr Djindjic urged Mr Perisic to resign, he has also demanded the sacking of the military authorities responsible for the arrest.
The Kostunica-Djindjic power struggle has simmered for months.
Mr Kostunica is a moderate nationalist, while the younger Mr Djindjic prefers to be seen as a pro-Western liberal.
Washington said last week it had received an apology from Yugoslavia's foreign minister, an ally of Mr Djindjic, for the treatment in custody of Mr Neighbor.
The US embassy says he was beaten, interrogated and held for 17 hours.