It is now a category four storm as it powers across the Gulf of Mexico. It is due to make landfall on Texas’ central coast very late on Friday.
Greg Abbott has asked for more federal aid to cope with 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 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”. 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.