On the website, users can enter personal details, the desired date of the test and whether they want a positive or negative result.
Both types of result can be used on Denmark’s now-active corona passport system.
Once details have been entered the website produces a printout signed by Falck doctors.
“It has come to our attention that false documentation of Covid-19 tests featuring our logo is being spread,” the company’s head of press communications Lisbeth Nedergaard said.
“We have therefore decided to file a police complaint against the site on which our logo appears,” Nedergaard added.
Meanwhile, public health platform sundhed.dk has also filed a police complaint against a website which offered instructions for producing fake corona passports featuring PCR tests.
The identities of the websites in question has not been released, but they are not thought to be the same website.
The Justice Ministry has previously confirmed to broadcaster DR that anyone caught breaking the rules in this way could face up to four years in prison.
That is because the normal sentence of around two years in prison for fraudulent use of documents can be double under a special, recently-introduced clause in the law providing for double sentences for crimes committed in relation to the Covid-19 pandemic.