According to the BBC, the woman has been identified by friends as Kim Wall, a freelance journalist based in Beijing who travels extensively.
Mr Madsen is due to appear before a judge on Saturday. He is "willing to co-operate" and hasn't decided whether to appeal the detention ruling, Hald Engmark said.
In a statement to Swedish paper Aftonbladet, Ms Wall's family said: "It is with great dismay we received the message that Kim is missing".
The 30-year-old Sweden-born freelance journalist had studied at the Sorbonne university in Paris, the London School of Economics and at Columbia University in NY where she graduated with a master's degree in journalism in 2013.
FILE - This is a April 30 2008 file photo of submarine owner Peter Madsen.More news: Abbasi arrives in Karachi on maiden trip as prime minister
Prosecutor Louise Pedersen said Madsen faces the preliminary manslaughter charge "for having killed in an unknown way and in an unknown place Kim Isabell Frerika Wall of Sweden sometime after Thursday 5 p.m".
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".
Kristian Isbak, who had responded to the Navy's call to help locate the ship, sailed out immediately Friday and saw Madsen standing wearing his trademark military fatigues in the submarine's tower while it was still afloat.
While the UC3 Nautilus was going down, Isbak says "came up again, stayed in the tower until water came into it as the submarine started sinking".
The Danish owner of a homemade submarine that sank south of Copenhagen has been arrested on suspicion of murdering a Swedish journalist who was a passenger on the vessel.More news: Usain Bolt, India's relay teams in action
Peter Madsen was arrested Friday on preliminary manslaughter charges after his 40-ton, almost 18-meter-long (60-foot-long) UC3 Nautilus sank.
They described her as "160cm tall, 56kg, 30 years old, with reddish-brown hair and green eyes".
However, when she failed to return home later that day, her anxious boyfriend contacted the authorities, which led to a full-scale search for the submarine in the early hours. Half an hour later "the submarine suddenly sank and the owner was subsequently rescued on a private motorboat sailing him in port".
Copenhagen Deputy Police Inspector Jens Moller Jensen says investigators are looking for witnesses who may have seen the woman after the time Madsen reported she disembarked.More news: Wunderkind Shapovalov blasts into Rogers Cup semifinals
- Premier League table 2017/18
- City's mega-spending is a one-off, insists Guardiola
- Political tensions continue to weigh on world stocks
- Zillow Group, Inc. (NASDAQ:Z) Rating Reiterated by Cowen and Company
- Biggest Threat to Trump's Agenda Comes from Fellow Republicans
- Sir Mo Farah claims silver in 5000 metres at world athletics championships
- Arsenal to make big Alexis Sanchez push
- Anna Paquin jokes about BBC breast broadcast
- The Spider-Man Spinoff Silver And Black Is Coming In February 2019
- North Korea refuses to stop nuclear weapon development, threatens US over sanctions