The agency said Lieutenant General Emadeddin Adawi was "sacked" and the job of military chief of staff given to Lieutenant General Kamal Abdul Maaruf, who was previously the army's inspector general.
"Major changes in the leadership of the armed forces and Lieutenant General Kamal Abdul Maaruf military is (the new) chief of staff," SUNA said.
It gave no reason for the shake-up, which comes just two weeks after Bashir replaced powerful intelligence chief Mohammad Atta.
Atta was replaced by Salah Abdallah Mohammed Salih, a former head of the powerful National Intelligence and Security Service (NISS).