The latest NRS report, which covers September 7 to 13, shows two of those deaths occurred in care homes while three were in hospital.
It comes as First Minister Nicola Sturgeon revealed today there have been 267 further positive tests for the virus across the country, including 105 in the Greater Glasgow and Clyde health board area and 45 in Lothian.
Ms Sturgeon also said there has been one more death recorded in the latest 24 hour period.
It means the number of people who have died after testing positive for Covid-19 is now 2,501.
But the NRS measure, which includes deaths of people where Covid-19 was mentioned on the death certificate, stands at 4,236 as of Sunday. This report will not include the two deaths announced by Ms Sturgeon on Tuesday and Wednesday this week.
The latest NRS weekly figure of five deaths is an increase of three from the previous week.
However the latest report says that, since mid-July, weekly deaths involving Covid-19 have remained around the same level with some weekly fluctuation.
Data suggests link to deprivation
The report also said that, after adjusting for age, people in the most deprived areas were 2.1 times more likely to die with Covid-19 than those living in the least deprived areas.
People living in larger urban areas are also more than four times more likely to die with Covid-19 than those in remote rural locations.
Of those who died with Covid-19 in March to August, 92 percent had at least one pre-existing condition, the most common of which was dementia and Alzheimer’s disease followed by ischaemic heart disease.
Alan Ferrier, head of demographic statistics, said: “Every death from this virus represents heartbreak for families and communities across the country who have lost loved ones.
“However, since mid-July the number of deaths involving COVID-19 have remained relatively low, averaging out at one death every other day.
“The updated analysis once again shows that COVID-19 mortality rates are higher in urban, more populated areas, and in areas of highest deprivation.”