According to the information, the registration should be for preventive purposes, in the event that some countries would send more migrants on to Germany.
The information comes from sources within the governing party SPD and the Ministry of the Interior, the newspaper writes.
Interior Minister Nancy Faeser (SPD) will meet colleagues from the three neighboring countries on Monday to discuss the issue.
Within the Schengen area, countries must notify the EU of decisions on border controls. Currently, Germany only has fixed controls at the border between Bavaria and Austria. They were introduced in 2015 and have since been extended every six months.