First launched in 2008, the ship was famous for being the world's biggest privately owned submarine, but it had sat in disrepair for seven years until April.
Police have identified the missing journalist as Kim Wall, a 30-year-old Swedish woman who was writing a feature about Madsen.
Hald Engmark says Madsen is "willing to cooperate" and hasn't decided whether to appeal the detention ruling.
Madsen, 46, was charged with the manslaughter of Kim Wall on the same day he was rescued from the sea by the Danish navy.
Madsen said he had left her on the end of the Refshaleøen island at around 2030 GMT, according to police.
Mr Madsen appeared calm during an interview with Danish television shortly after the submarine sank, saying: "I am fine, but sad because Nautilus went down".
Kim Wall's family said in statement emailed to The Associated Press on Saturday, "It is with great dismay that we received the news that Kim went missing during an assignment in Denmark".
She lived in New York and Peking, her family said, and had written for The New York Times, The Guardian, the South China Morning Post and Vice Magazine, among other publications.
The journalist was born in Sweden and studied at the London School of Economics, Columbia University in NY and the Sorbonne in Paris.
It is believed a search will be carried out once the vessel has been towed to port later today.
Before the submarine's relaunch, Madsen had been locked in a fierce ownership dispute with a board representing the group of volunteers who helped him build it.
'He then climbed down inside the submarine and there was then some kind of air flow coming up and the submarine started to sink, ' Isbak told The Associated Press.
Madsen's actions are being scrutinized because he is being detained in the disappearance of a Swedish journalist who had been onboard his submarine Thursday night.
The 40-ton, 18-meter long Nautilus, one of three subs built by Madsen, is now sitting under 7m (24ft) of water with divers unable to enter it safely.
However, Swedish police said later on Friday they were investigating the whereabouts of a missing woman who had been on the submarine at some point. He has denied responsibility for the 30-year-old woman's fate, saying she disembarked earlier.
Ms Wall's boyfriend had called out the authorities after the submarine failed to return after she had boarded it on Thursday. The Navy said it was spotted sailing but then sank.
Miss Wall, 30, has not been seen since the ill-fated voyage, though Madsen claims he dropped her off on an island in Copenhagen's harbor not long into the voyage.