Anne-Elisabeth Falkevik Hagen, 68, married to a real estate and energy magnate, disappeared on October 31, police said.
Detectives had been investigating the case discreetly for several weeks but decided to make it public on Wednesday in the hope that someone would come forward with information.
"A ransom demand and serious threats have been issued," Inspector Tommy Broske told reporters.
"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.
The authorities had received no sign of life from the woman, "but we haven't received any indication that she isn't alive either", Broske said.
Investigators said they had no suspects at this stage.
According to daily Verdens Gang, the ransom demand was for nine million euros ($10.3 million) in the cryptocurrency Monero. Cryptocurrencies are hard to trace.
Police refused to confirm the sum but said they had advised the family not to pay.
Such incidents are extremely rare in the wealthy Scandinavian country, which enjoys a generally low crime rate.
"We have not known anything comparable in this country," said Broske.
Broske said investigators had had "very limited" contacts with the self-described kidnappers online, but refused to disclose the date of the most recent contact.
Norway's 172nd richest
A housewife, Hagen is married to Tom Hagen, also 68 and Norway's 172nd richest man, according to the magazine Kapital, with an estimated fortune of 1.7 billion kroner ($200 million).
The real estate investor owns 70 percent of electricity company Elkraft, which he co-founded in 1992, Norwegian media reported.
The parents of three adult children, the couple lived a quiet, retiring life in an elegant but understated home, now blocked off by police tape.
While he keeps a low profile, Tom Hagen has been the subject of news articles about his business success.
A lawyer for the family said they had not paid the ransom.
"The family decided to follow the police advice," Svein Holden told reporters.
"There's no doubt that for the family, Anne-Elisabeth's kidnapping is a horrible and inhumane act," he said, expressing hopes that news of her disappearance would lead to some kind of sign that she is alive.
According to VG, a note written in poor Norwegian was left at Hagen's home warning that she would die if police were alerted.
Investigators refused to comment on that report, but said international police were cooperating on the case.