The comments came hours after the Swedish king said the resignation of three members from the Swedish Academy was "deeply unfortunate and risked seriously damaging" the body's "important activities."
After a Nobel board meeting Wednesday, the foundation issued a statement noting that "trust in the Swedish Academy has been seriously damaged" and demanding that the group take specific actions to restore that trust.
The Nobel board said that included making sure the Swedish Academy can carry out the 2018 Nobel selection "in a credible manner," resolving the group's conflicts of interest and confidentiality issues according to Nobel regulations and referring any "suspected criminal acts" to law enforcement officials.
The prestigious academy has been in turmoil after the resignations Friday of members Klas Ostergren, Kjell Espmark and Peter Englund, who quit after the secretive 18-member academy voted against removing a female colleague whose husband is embroiled in sexual misconduct allegations.
Academy members are appointed for life and resignations are extremely rare. In gender-neutral Sweden, it was also a surprise that three male cultural figures would demand that a woman be ousted from a Nobel-awarding committee for her husband's alleged actions.
The men were calling for the removal of academy member Katarina Frostenson, whose husband, Jean-Claude Arnault, has been accused of sexual misconduct and leaking Nobel winners' names.
Arnault was banned in December by the Swedish Academy from attending a Nobel banquet after the Swedish daily Dagens Nyheter published allegations from 18 women claiming to have been assaulted or raped by Arnault.
Arnault, the head of a Stockholm culture center, denies the alleged assaults, which reportedly occurred between 1996 and 2017.
In March, Swedish prosecutors said an investigation into reported rape and sexual abuse by Arnault during 2013 to 2015 had been dropped, but a probe into other criminal acts would continue. It did not elaborate.
Separately on Tuesday, one of Sweden's biggest dailies, Dagens Nyheter, cited an internal Academy probe alleging that Arnault leaked the names of winners the Nobel Literature Prize seven times, starting in 1996. The Dagens Nyheter said the names include U.S. singer-songwriter Bob Dylan and Belarus journalist Svetlana Alexievich.
It was not clear how he had learned who the winners were. The academy never discusses potential winners and has kept nominations secret for half a century.
In his second statement this week, Swedish King Carl XVI Gustav said "it is crucial that all involved now realize their responsibility for the institution and contribute to resolving conflicts."
The king is the academy's patron, and must approve any of the body's secret votes before they are made public. The king said he will consider new rules for members to withdraw.
The Academy itself has declined to comment on the turmoil. It says it doesn't want to take the limelight from the Nordic Prize 2018 - another award it hands out - which was given Wednesday to Agneta Pleijel, a Swedish novelist, poet, playwright, journalist and literary critic.
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