It was centred about 235 miles (380 kilometres) southwest of Manzanillo, Mexico, and was moving northwest at 10 mph (17 kph).
The National Hurricane Center said the chances of a tropical depression forming over the next five days were low - 20 percent.
The hurricane is moving north-west at about 10mph - a course it is expected to take until mid-week, the NHC said. The NHC noted that some additional strengthening is possible Monday but a decline in strength is expected to start by Tuesday.
At Category 3, Bud was classified as a major hurricane.
"On the forecast track, the core of Bud and its stronger winds are expected to remain well offshore of the southwestern coast of mainland Mexico". Rainfall of 3 to 6 inches, with isolated patches of 10 inches, was possible into Tuesday afternoon.
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.
"Swells generated by Bud will continue to affect portions of the coast of southwestern Mexico during the next few days", the NHC said.
The Atlantic hurricane season lasts until November 30.
Aletta was the first hurricane of the season and went from a tropical storm to category 4 hurricane in just 24 hours last week but it too was located well away from the coast.