The deceased were identified as Jakir Hossain, 45, a resident of Sadar upazila in Patuakhali, and Niru Begum, 45, wife of Md Dulal of Puranpara in the city.
It is unclear whether their samples have been collected for COVID-19 tests.
Niru was the first one to die at the Corona Unit of SBMC in the early hours of Sunday. Jakir's death was reported about eight hours later.
SBMC Director Dr Md Bakir Hossain said Niru's relatives rushed her to emergency around 11:45pm.
He suggested them to take her to Corona Unit of the hospital after hearing her symptoms. She died around 12:05am.
She had been suffering from diarrhea and underwent treatment at Barisal General Hospital around three days ago.
She returned home after recovery but fell sick again with fever, pain in the throat and asthma. She also had diabetes and high blood pressure, the doctor said quoting relatives.
The relatives took the body home soon after her death.
Dr Hossain said the relatives of Jakir Hossain Took him to the medicine department of the hospital first on Sunday afternoon at 5:50pm.
He was later shifted to Corona Unit of the hospital where he died around 7:20am.
Also read:Bangladesh reports first coronavirus death
The Institute of Epidemiology, Disease Control and Research (IEDCR) was informed about the cases and the authorities permitted burial, the doctor added.
Jakir had been suffering from asthma for a long time, his family sources sad.
Six patients with symptoms of coronavirus were admitted to SBMC since last night. Doctors said the hospital lacks any equipment to test coronavirus.
Coronavirus, first reported in China in December last year, is affecting 199 countries and territories around the world and two international conveyances.
It has infected 663,741 people globally and killed 30,880 of them so far -- a mortality rate of 18 percent, according to Worldometer. But 142,183 of the patients have recovered while 490,678 are undergoing treatment. Of them, 25,207 patients are in critical condition.
Bangladesh has so far confirmed 48 COVID-19 cases and five deaths. The country has tested 1,068 samples till date.