Düsseldorf-based daily Rheinische Post cited government sources saying the conservative Christian Democrat had already taken the offer to the Finance Ministry on Monday in an effort to contribute to Berlin's budgetary consolidation efforts.
Such a blanket cut could save some €200 million per year, the paper said.
The decrease would mean that the new minimum benefits would be €230 per month and the top rate would reach €1,730 per month for working parents taking time off after having a baby.
According to the paper, the Finance Ministry, headed by Köhler's fellow party member Wolfgang Schäuble, welcomed the offer of savings.
A decision on the offer will be made in the coming weeks, the paper said.
Current parental leave benefits pay new mothers or fathers 67 percent of their net salary if they decide take time off or reduce their weekly schedules to a maximum of 30 hours per week.
They can receive a maximum of €1,800, and a minimum of €300 for up to 14 months through the programme. However, in a bid to get fathers more involved, one parent may take a maximum of just 12 months of leave, earning the next two months of Elterngeld only if the other parent cares for the child during that time. Single parents get the full 14 months of compensation.