The kidnappers have threatened to use violence against her if her family does not meet their demands, Norway's Aftenposten newspaper reported today.
Chief investigator Tommy Broeske said 68-year-old Anne-Elisabeth Falkevik Hagen has been missing since October 31.
In a statement, police said they had made the decision to make the case public despite threats from the kidnappers, in order to appeal for more information.
Norway's financial magazine Kapital named him 172nd on its list of the country's wealthiest people.
Mrs Hagen is married to Tom Hagen, a businessman with an estimated fortune of 1.7bn Norwegian krone (€174m; £156m) made in the real estate and energy industries.
Sources have now indicated that a written message was found in the Hagen house, which demanded a $10 million ransom be paid entirely with the privacy-focused cryptocurrency Monero.
The house (C) of Norwegian real estate investor Tom Hagen and his wife Anne-Elisabeth Falkevik Hagen, who according to the authorities is suspected to have been kidnapped, in Fjellhamar, Norway, Jan. 9, 2019.
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'As the case now stands, police have advised the family not to settle the claim.
There has been no evidence since she disappeared that Mrs Hagen is safe, police say. Mr Broeske declined to comment other than saying "the threats (in the note) were of a very serious character".
Police fear the wife of one of Norway's richest men is being held ransom for an eye-watering 9 million euros ($10M) - payable only in the popular cryptocurrency Monero.
Police have been investigating the case discreetly for several weeks but decided to make it public in the hope that someone would come forward with information, Broske said.
Broeske said "those behind have chosen to communicate digitally and we have had no other type of contact". Mr Broeske said they were working with Europol and Interpol on the case.
"Our main theory is that the victim was kidnapped by unidentified perpetrators at her home" in Lorenskog, about 20 kilometres (12 miles) from Oslo, he said.