Norbert was about 215 miles (345 km) from the southernmost point of Baja California shortly before 0000 GMT and the hurricane was expected to turn northwest later in the evening, the US National Hurricane Center (NHC) said.
That would mean Norbert hits the Mexican coast on Thursday, the NHC said, bringing tropical storm conditions to the region and 3 to 5 inches of rain on Friday.
Norbert, which was blowing maximum sustained winds of almost 75 miles per hour (120 km per hour), is expected to strengthen during the next day or so, the Miami-based NHC added.
Mexico's state-run oil company Pemex does not have any major installations close to the affected region.