"It is with a great concern that we, her family, received the news that Kim is missing after an interview with Peter Madsen in Denmark", the statement said.
The woman's partner raised the alarm early on Friday when the vessel did not return from what was supposed to be a short journey.
Footage aired by Denmark's TV2 shows Mr Madsen getting out of what appeared to be a private boat, giving reporters a thumbs-up sign, saying: "I am fine, but sad because Nautilus went down". Denmark's navy says that Madsen's privately built submarine that had been feared missing in Danish waters has been found and the crew is safe.
On Saturday, the rescue vessel Vina recovered the UC3 Nautilus submarine, which is being treated as a possible crime scene, from the bottom of the sea near the island of Dragoer south of Copenhagen. The inventor has denied wrongdoing and said that he dropped Wall-who was reported to be the only other person on board the submarine with him-off on an island close to Copenhagen Thursday night before the submarine sank, according to AP reports.
Mr Madsen was rescued, but there was no trace of the woman.
"So far it hasn't been possible to see what there is inside the submarine", it said.
The police in Sweden said they had tried without success to contact the woman by phone. Nothing has been heard from her since Thursday evening. "But I guess that was pretty good, because I otherwise still would have been down there". The navy says that the 40-ton, almost 18-meter-long (60-foot-long) submarine with at least two people on board had been "found sailing" south of Copenhagen.
Media frenzy surrounding the case forced the hearing to be moved to a larger courtroom in Copenhagen on Saturday.
Madsen describes himself as an "inventrepaneur" on the website for his Copenhagen-based company.
"My passion is finding ways to travel to worlds beyond the well-known", Madsen wrote on the site.
"The undersea world is close, it's lovely, always just around the corner in Denmark".
The Nautilus set out from Refshale Island, a former shipyard redeveloped as an entertainment and activity spot, at about 7 p.m. local time on Thursday, according to a police timeline. It was expected to arrive in the Danish capital by the end of the evening, said a police statement. The Navy said that the sub had been spotted sailing but then sank shortly afterward.