They all had underlying health conditions while living at Burlington Court in North Lankarshire, Glasgow.
None of the fatalities were tested for Covid-19 as they were not admitted to hospital, but health officials have told their families of a suspected link to Covid-19.
Two staff members at the care home are being treated in hospital after contracting the disease.
Four Seasons Health Care, which runs 90-bed home, said it had "strict protocols on infectious diseases" in place and is “closely monitoring” the health of other residents.
A spokeswoman added: “With deep sadness, we can confirm that 13 residents at Burlington Care Home have passed away over the past seven days.
“Our condolences are with their families and we are providing them with our ongoing support during this difficult time.
"The passing of a loved one is always traumatic irrespective of the circumstances.”
The Care Inspectorate, the regulator, said it is aware of the “tragic death of residents at this care home as a result of suspected Covid-19” and is liaising with Burlington Court.
It comes amid mounting concern at shortages of personal protective equipment (PPE) in the care sector.
NHS staff have crowdfunded hundreds of thousands of pounds to buy more masks, visors and gloves.
Meanwhile, the Doctors' Association chief has warned doctors and nurses could quit if supplies are not bolstered.