The newspaper said Economy Minister Kaoru Yosano will present the estimate of 15 trillion yen to 25 trillion yen ($185 billion to $300 billion) at a Cabinet meeting on Wednesday.
The March 11 magnitude-9.0 quake and tsunami devastated Japan's northeastern coast and triggered a crisis at a nuclear power plant.
Utilities have imposed power rationing, many factories remain closed and key rail lines are impassable.