Sudan's second-largest city "witnessed riots and illegal gatherings" on Wednesday, SUNA said, amid weeks of demonstrations.
SUNA quoted police as saying that they knew of three deaths and several people being wounded and that these attacks were being investigated.
No other details were immediately available.
Bashir vowed at a rally of thousands of supporters in the capital Khartoum on Wednesday that he would stay in power.
His speech failed to quell the unrest, with security forces fighting running battles on Wednesday with protesters in Omdurman on the other side of the Nile to the capital.
Protesters have been staging demonstrations almost daily for weeks, enraged by shortages of bread and foreign currency. The unrest has come as the ruling party has pressed ahead with plans to change the constitution so Bashir can stay in office beyond his present term, which ends in 2020.