Harvey, currently a category three storm in the Gulf of Mexico, will make landfall on Texas’ central coast late on Friday or early on Saturday.
Greg Abbott has asked for more federal aid for the storm, which may be the worst to hit the US in 12 years.
The National Hurricane Center said storm surges may bring life-threatening floods in and around Houston.
The storm may bring 35in (97cm) of rain, 130mph (210km/h) winds and 12ft (3.5m) storm surges, say forecasters.
Mr Abbott said Harvey was now “turning into a very complex and dangerous hurricane”. It is likely to strike at the heart of Texas’s oil refining industry, with Corpus Christi, a port city of 320,000 people, in the path of the storm.
In a letter to President Donald Trump requesting federal aid, Mr Abbott said: “The storm surge, coupled with the deluge of rain, could easily lead to billions of dollars of property damage and almost certainly loss of life.”
The White House said Mr Trump is likely to visit Texas early next week.
While Harvey is packing strong and potentially devastating winds, the biggest threats to Texas are rainfall and the storm surge.
Meteorologists are warning of extremely high volumes of rainfall as the storm stalls over the middle Texas coast.
They say Harvey could remain in the area, dumping rain until the middle of next week.
Oil-rich Houston, the fourth biggest city in the US, could face up to 20in of rain over the coming days, officials there said.
On top of that, the central Texas coast is likely to see a significant storm surge – this happens when low pressure at sea “lifts” tides to a level higher than normal, up to 12ft in this case, and high winds then blow the water inland.
The NHC said it expected “catastrophic flooding” across the coast and in some inland areas throughout south-east Texas.
Hours before Hurricane Harvey was expected to make landfall, the governor and the mayor of Houston seemed at odds about whether to evacuate the nation’s fourth-largest city.
In recent days, Houston officials have recommended that most residents wait out the storm in the flood-prone city. But on Friday, Gov. Greg Abbott abruptly urged a different approach.
“I would urge people to strongly consider the evacuation process,” Mr. Abbott said during a news conference. “There is a possibility, the probability, that a lot of people will go for a long time without the basic necessities.”
Mayor Sylvester Turner, in a series of tweets after Mr. Abbott’s remarks, tried to rebut the governor and calm his city.
“Please think twice before trying to leave Houston en masse,” Mr. Turner posted on Friday afternoon. “No evacuation orders have been issued for the city.”
Seven coastal counties from Corpus Christi to the western end of Galveston ordered mandatory evacuations of at least some areas. Mayor Joe McComb of Corpus Christi encouraged residents to leave voluntarily.
Houston, with a population of about 2.3 million people, has a history of floodwaters that led to fatalities. In 2001, Tropical Storm Allison pummeled southeast Texas and was blamed for 22 deaths in the Houston area.
Hurricane Rita, in 2005, never struck Houston directly, but the evacuation, spurred by the recent horror of Katrina in New Orleans, caused disastrous traffic jams; more than 100 people died while trying to get out of town.
Mr. Abbott described the storm as a “very complex and dangerous hurricane” that will hammer Texans not only with high winds but “record-setting flooding in multiple regions of the state.” The storm is also expected to spawn multiple tornadoes.