Out of 2,600 randomly selected people, 1,071 had so far agreed to be tested for antibodies.
Only 12 of those tested positive, corresponding to a rate of about 1.1 percent.
SSI cautioned that the results were preliminary and there were several factors that made it difficult to say whether the results were indicative of the entire Danish population.
They had only been able to carry out tests at five locations, meaning they had been limited to people living in those, or neighbouring, municipalities.
“Furthermore, whether the figures can be applied to the entire Danish population can also be affected by whether groups with different patterns of infection choose or not choose to accept the offer to be tested,” Steen Ethelberg at SSI said in a statement.
Experts interviewed by broadcaster DR said the results were concerning and showed that the country was vulnerable to the spread of the virus picking up speed again.
“At the collective level we have no resilience, and that means there is a potential for epidemic spread again,” Jens Lundgren, professor of infectious diseases at one of Denmark's largest hospitals Rigshospitalet, told DR.
Denmark on Wednesday reported a total of 11,117 confirmed cases of Covid-19 and 554 deaths.
Meanwhile, the parties of the country's parliament agreed on the next phase of opening up the country after an initial lockdown period.
On April 15, the country started reopening pre-schools and resuming classes for the youngest primary school children – under strict social distancing and hygiene guidelines.
Danish middle schools followed suit this week and with the deal struck on Wednesday, museums, movie theatres and zoos among other things would also soon reopen.