Without citing a source, Bild reported that German Chancellor Angela Merkel and her interior minister had agreed the closures with the state premiers involved and that they would take place from 8 a.m. local time (0700 GMT) on Monday.
Goods would continue to flow between the countries and commuters could also still cross the borders, it said.
The aim was to contain the coronavirus and to stop panic buying by foreigners which has led to some supply problems in the border regions, Bild added.