Ms Borst, 81, helped push through legislation in 2001 that made the Netherlands the first country to legalise euthanasia.
Forensic scientists were unable to say why she died. Police have said it was either an accident or a crime.
Ms Borst, who was a medical academic, served as minister from 1994-2002.
Her body was found on Monday evening and the area around her home near the central city of Utrecht was cordoned off.
Initial inquiries proved inconclusive, however police on Tuesday ruled out death by natural causes.
A full post mortem examination will now take place at the Netherlands Forensic Institute, they say.
"She may have had an accident after she felt unwell, but that cannot yet be said with certainty," a police spokesman told Dutch media. "So we cannot yet say whether any crime may have been involved."
A spokesman had earlier told Dutch radio there were no indications of anything untoward happening.
Els Borst had appeared at a conference of her liberal D66 party on Saturday and her death came as a shock to political colleagues in The Hague.
For a time, she led her party in the 1990s and Prime Minister Mark Rutte said her death was a great loss for Dutch politics.
"She ensured a breakthrough in the field of euthanasia for which very many people are still grateful," said Health Minister Edith Schippers.
She was also behind legislation governing the use of tissue from aborted foetuses for medical research.
Dutch authorities have begun an investigation into the death of former health minister Els Borst, whose body was found in her garage by a friend.