Texas Gov. Greg Abbott said Sunday the federal government's response and assistance to areas hit by Hurricane Harvey thus far has been on time and well-coordinated.

"We could not be more appreciative of what the federal government has done, from the president on down," Abbott told ABC's George Stephanopoulos.

"Because, everything we have asked for, they have given us. The most important thing was that I made a disaster declaration that the president granted very swiftly. What that does, it triggers [the Federal Emergency Management Agency's] involvement. FEMA has been actively involved and engaged in this whole process long before the hurricane even hit ground. And because of their assistance it means that Texas will be able to begin the rebuilding process very swiftly."

Abbott said he expects Texas will be adding to the number of counties in the disaster declaration as the flood waters continue to swell, even though Harvey has been downgraded from a hurricane to a tropical storm.

The National Weather Service has gone so far as to issue an alert telling homeowners not to go up to their attics to avoid flood waters, but to go up on to the home's roof instead.

President Trump indicated Sunday he wanted to visit the region as quickly as possible, and Abbott said that would likely all depend on where Trump decided to go.

"We're already, for example, involved in the cleanup process in Corpus Christi. If the president were to visit there, it wouldn't hinder any efforts," he said.

Abbott told "Fox News Sunday" it's too early to make a detailed damage assessment or to say if the death toll from the storm will rise, but he's warning residents to brace for a tough week.

"It is bad and growing worse," he said, of the flooding situation in Houston and surrounding areas.

"We've got a big task on our hands for the next several days, if not a week," he added.

Harvey has stalled over the Houston area and dumped about 20 inches of rain between Saturday night and Sunday morning. Forecasts show an additional 25 inches of rain could fall on some areas.

Abbott said the state has to deal with a unique disaster situation, due to the wind damage that destroyed homes and other buildings along the coastline and the flooding in Houston.

However, he's hoping many people listened to warnings to evacuate the area and the death toll may not be catastrophic.

"There were a lot of warnings to evacuate, there was a large amount of evacuations from that area and a lot of warnings that probably saved a lot of lives," he said.