The Arab African country, which has reported 4,146 coronavirus cases and 184 deaths, released more than 4,000 prisoners in March as a precaution against the spread of the disease in jails.
But political sensitivities mean all those jailed for offence committed as part of the administration run by the veteran Islamist, who was ousted in an uprising a year ago, remain in custody.
They include Ahmed Haroun and Abdel Raheem Muhammad Hussein, both wanted by the International Criminal Court (ICC) for alleged crimes in Sudan’s Darfur region, and who the prosecution service said had tested positive for coronavirus.
Haroun was a senior Bashir aide while Hussein served as defence and interior minister. Ali Osman Taha, a former vice president, was also infected, it said in the statement said.
All three are under health quarantine in Khartoum hospitals, the statement said.
Bashir, scores of politicians close to him and members of his family, were detained after the uprising on various charges including corruption and violence.
COVID-19 tests on two other detainees including Ali al-Bashir, a brother of the former president, came back negative, though they were quarantined as a precautionary measure.
Tests results are still awaited on two other suspected cases, the prosecution service said, adding that all other detained members of the former regime had refused to be tested for coronavirus.
This month, Sudan, which has a poor health system, extended a lockdown of the state of Khartoum by two weeks to slow the spread of the virus.