Kumar, who has been in Japan since Monday, mooted the proposal during a meeting with Nara Governor Shogo Arai who supported the idea "and agreed to follow it up", an official release said here.
The Nara Governor also praised Kumar for his "visionary leadership" and the progress witnessed by Bihar during his tenure as Chief Minister.
The Nara prefecture is home to many renowned shrines, including the Todai-ji Buddhist temple complex which has been declared a World Heritage site by UNESCO.
The two leaders also spoke about Bodhgaya, the pilgrim town where Buddha is said to have been enlightened some 2500 years ago, and the common cultural roots shared by between Nara and Bihar.
Indian ambassador to Japan Sujan Chinoy was present during the meeting and the two leaders were told that the embassy "will provide its support" to the proposed "sister state" agreement.