Bergwall, previously known as Thomas Quick, claimed responsibility for and was found guilty of eight murders before withdrawing his confessions and being cleared of all charges.
Säter criminal psychiatric clinic, in Falun, central Sweden, has applied to an administrative court for Bergwall to be transferred to open psychiatric care, according to local newspaper Dalarnas Tidning.
The clinic had previously refused to consider releasing Bergwall.
Bergwall has been held at the Säter psychiatric hospital since 1991, after he was convicted of robbing a bank.
Säter's chief medical officer must now submit a plan on how open care would work in Bergwall's case, and the court will then rule on what form this care should take.
In October last year, an administrative court reviewed whether Bergwall should be released from Säter, and if so how, although no final decision was reached.
Bergwall's lawyer, Thomas Olsson, said at the time of the hearing that there had not been a legal basis for holding Bergwall in custody since 2002.
Speaking following today's developments, Olsson said his client welcomed the news.
"He's delighted; this is the end of an extremly long process that has taken many years. He can now look forward to starting a new life in which he can live in freedom," said Olsson.