It was centered about 440 miles south-southeast of the tip of the Baja peninsula and was moving northwest at 7 mph.
The hurricane is moving north-west at about 10mph - a course it is expected to take until mid-week, the NHC said.
At Category 3, Bud was classified as a major hurricane.
"Some additional strengthening is possible today, but a slow weakening trend is expected to begin by early Tuesday".
Some 3 to 6 inches of rain accumulations are expected in much of southwestern Mexico, with isolated maximum amounts of 10 inches, the NHC said in the advisory.
NOAASatellite image shows Hurricane Bud
Even if no tropical system develops, forecasters warned that the system could dump heavy rain on flood prone areas in Nicaragua, Honduras and other parts of Central America as well as the Yucatan through Thursday.
The swell also will generate strong rip and side currents, creating hazardous conditions for swimmers and surfers.
A tropical storm watch is in effect from Manzanillo to Cabo Correintes.
The National Hurricane Center is monitoring an area of storms over the southwestern Caribbean Sea.
The center said the hurricane's core was moving farther from Mexico's southwestern coast but still could generate risky heavy surf and rip currents over the coming days. Aletta surged from a tropical storm to a Category 4 hurricane late last week over the span of just 24 hours.
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Trump hired him in April to replace Gary Cohn as the head of the National Economic Council, reportedly because he enjoyed Mr. Associated Press writers Zeke Miller in Singapore and Jill Colvin in Washington contributed reporting .