The draft law submitted by equality minister Jana Schiedek was agreed by the state government on Tuesday to increase the influence of women.
Over the last few years the share of women in committee positions was just over 30 percent, the Hamburger Abendblatt newspaper reported.
Supervisory boards of publicly owned companies had a share of around 25 percent, the paper said.
"Also in Hamburg there are decidedly too few women in supervisory, decision-making and advisory boards. This is not because there is a shortage of brilliantly qualified women, but they are not sufficiently represented in the relevant committees," Schiedek said.
Only a binding quota would push those concerned to consider those women, she suggested.
"In order to achieve a lasting and sustainable increase in the share of women in such positions, we need a legal quota," she added.
She said women would in the future take their rightful places in influential places, "from the supervisory board of the Hamburg public transport system to the art commission."
Chancellor Angela Merkel recently defeated a parliamentary attempt to force her hand on the matter nationally, with a motion to implement a 40-percent quota for women in supervisory and management boards of large companies within a decade.
Although the opposition failed in its attempt to force through the change, such was the pressure on Merkel from within her own party that she agreed to include a pledge for a 30-percent quota to be introduced by 2020 into the party manifesto.
Merkel has long opposed the idea of a legal quota, both in Germany and in the European Union as a whole.