The hangar will be built near the state-run museum as a temporary climate-controlled home to make Endeavour available for public viewing within weeks of its arrival in the fall, the Los Angeles Times reported Monday.
The museum will then begin construction of a permanent Air and Space Center with Endeavour as its centerpiece, officials said.
Endeavour, now at Cape Canaveral in Florida, being prepared for display, will be ferried to Los Angeles International Airport.
To move it to the science center, crews will have to dismantle traffic signals, cut trees and close roads on its route through the city, the Times reported.
Three others shuttle from NASA's fleet will be displayed in museums in Washington, New York and Florida.