Mortaza, 36, is also a member of Bangladesh's national parliament.
He has taken 390 wickets in 310 games across all formats since his international debut in 2001.
Spinner Nazmul Islam, 28, and batsman Nafees Iqbal, 34, who last played for the Tigers in 2018 and 2006 respectively, also tested positive.
Nafees is the elder brother of opener Tamim Iqbal, who succeeded Mortaza as one-day international captain earlier this year.
Coronavirus cases in Bangladesh now exceed 100,000, with more than 1,400 people having died.