Italy s La Nazione daily publication reported the first and second officers had tested positive for Covid-19.
No autopsy has been carried out on the master, who had tested negative for coronavirus in Italy and South Africa.
Italian public prosecutor Antonio Patrono has received a complaint from family members who alleged a lack of assistance from Indonesian authorities.
He has ordered an autopsy and asked Italian police to repatriate the body.
Served with distinction The ship is ready to leave; we await information from the competent authorities, operator Italia Marittima said in a statement.
The company expressed its deepest condolences to the family of the late captain, who the company said served the organisation for many years with professionalism and dedication.