President Reagan is likely to sign an order during Prime Minister Noboru Takeshita's visit to the United States this week that would permit the export of Alaskan natural gas to Japan, an aide to Sen. Frank Murkowski (R-Alaska) said. The aide said the White House Economic Policy Council approved the export plan last week. Japan has expressed interest in buying the Alaskan North Slope gas, the aide, Dennis Fradley, said. Taiwan and South Korea are also said to be interested. Shipments are unlikely to start until the middle of the next decade, however. Fradley said Japan should be eager for the gas to help it diversify its energy sources and lessen its dependence on oil from the Middle East. Murkowski estimated in a letter to Treasury Secretary James A. Baker III that the gas pipeline and export project could cost up to $10 billion.