Line Oma, a deputy municipal mayor, became the first person in the party to demand Giske resign from his post in a highly critical message posted on Facebook on Friday, reports newspaper Dagens Nyheter.
Her statement has received the support of Jonas Bals, until recently a key advisor to party leader Jonas Gahr Støre, according to the newspaper.
Oma, who is a member of Labour's network for female members, accused senior party members of “not doing their job” over the issue.
“I know why it is so painfully quiet in the party today: it is because nobody has done their job… Jonas [Gahr Støre] has not done his job. Annike [Huitfeldt], leader of Labour's women's network, has not done her job… the rest of us in the party have not done our jobs. We have quietly accepted the elephant in the room.
“I am sick of bureaucratic chat, talk of power struggles and factions. This is about what should have been said a long time ago: Labour cannot have a deputy leader that or future party leader that makes these kind of errors,” she wrote.
Bals confirmed to Dagens Nyheter via text message that he agreed with Oma's view.
Gro Balas, a former councillor and vice-chair with the Labour party's women's network, also said to the newspaper that she felt Giske should resign.
“It would have been the right thing to do to regain calm. The limits for [appropriate] behaviour must be well-defined and clear, particularly when we are responsible for younger people. Why has Giske not laid all his cards on the table, to save the pain of the woman who feel bad and have not dared to speak up. And why has Støre not also demanded he do so?” Balas said.
Giske on Thursday apologised for previous inappropriate behaviour towards women.
“I think I must accept that I have not been thoughtful enough about my behaviour and role. That can mean when there are differences in seniority or age or in social situations with a lot of alcohol,” he said on NRK's Dagsrevyen programme.
“I apologise for acting in a manner that individual people have found to be inappropriate or uncomfortable. I must take responsibility for it. It is my fault. I am very upset about it,” he said on the programme.
Giske spoke to the broadcaster after it emerged that he had met with party leader Støre, who asked him to explain his conduct following reports of inappropriate behaviour.
“I did not know about the cases they brought up, but I remember the situations that were described. Regardless of the circumstances, I take it seriously as a whole,” he said to NRK.
Bergens Tidende reported on Friday that it had contacted 13 of Labour's 18 regional leaders. All of those who spoke to the newspaper said they did not currently wish Giske to resign, and expressed their confidence in Støre and the party hierarchy's ongoing management of the issue.