At 18:00 (23:00 GMT), the storm was 45 miles (70 km) east of Corpus Christi, with maximum sustained winds of 130mph (215 km/h), the NHC said.
The storm may bring 40in (102cm) of rain and 12ft (3.5m) storm surges, say forecasters.
Mr Abbott said Harvey was now “turning into a very complex and dangerous hurricane” and that 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.