The slow-moving storm has brought catastrophic flooding to Texas, killed at least nine people, led to mass evacuations and paralyzed Houston, the fourth most-populous US city.
It had also roiled energy markets and wrought damage estimated to be in the billions of dollars, with rebuilding likely to last beyond Trump's current four-year term in office.
"My administration is coordinating closely with state and local authorities in Texas and Louisiana to save lives, and we thank our first responders and all of those involved in their efforts,” Trump told reporters at the White House on Monday.
Trump was scheduled to arrive on Tuesday morning in Corpus Christi, near where Harvey came ashore on Friday as the most powerful hurricane to strike Texas in more than 50 years. The president will later go to the Texas capital Austin to meet state officials, receive briefings and tour the emergency operation center, the White House said.
Forecasters could only draw on a few comparisons to the storm, recalling Hurricane Katrina, which lay waste to New Orleans and killed 1,800 people in 2005.
The administration of then-President George W. Bush drew accusations that his response was slow and inadequate - criticism that dealt a serious blow to his presidency.
Flood damage in Texas from Hurricane Harvey may equal that from Katrina, one of the costliest natural disasters in US history, an insurance research group said on Sunday.
In Texas, thousands of National Guard troops, police officers, rescue workers and civilians raced in helicopters, boats and high-water trucks to rescue the thousands stranded in the flooding, which transformed streets into rivers and caused chest-high water build-ups in scores of neighborhoods.
In Cypress, Texas, Kayla Harvey, 26, was monitoring Facebook, finding where people were stuck and organizing friends with boats to go out and help. "This is just what we do for our community. We don't wait for someone to come and help we just go out and do it," she said.
Before Harvey, the last Category 4 hurricane to make landfall in Texas was Carla in 1961. It packed high winds and torrential rains that destroyed about 1,900 homes and nearly 1,000 businesses, the National Weather Service said.
Runways turned into lakes
Among the most recent deaths from Harvey was a man who drowned on Monday night while trying to swim across flooded Houston-area roads, the Houston Chronicle quoted the Montgomery County Constable's Office as saying.
The storm center was in the Gulf of Mexico, about 115 miles southeast of Houston on Monday morning and was likely to remain just off the coast of Texas through Tuesday night before moving inland over the northwestern Gulf of Mexico on Wednesday, according to the US National Hurricane Center.
The NHC said Harvey was not expected to further intensify before its center crosses the coast again within 36 hours or so.
Since coming ashore, Harvey has virtually stalled along the Texas coast, picking up warm water from the Gulf of Mexico and dumping torrential rain from San Antonio to Louisiana.
The Houston metro area has suffered some of the worst precipitation with certain areas expected to receive more than 50 inches (127 cm) of rain in a week, more than it typically receives for a year.
Harvey was expected to produce another 7 to 13 inches of rain through Thursday over parts of the upper Texas coast into southwestern Louisiana, the National Weather Service said.
"These stationary bands of tropical rain are very hard to time, very hard to place and are very unpredictable," said Alek Krautmann, a weather service meteorologist in Louisiana.
Schools and office buildings were closed throughout the Houston metropolitan area, where 6.8 million people live.
The Federal Emergency Management Agency director, Brock Long, estimated that 30,000 people would eventually be housed temporarily in shelters.
Houston and Dallas have set up shelters in convention centers and Austin was preparing to house as many as 7,000 evacuees.
Hundreds of Houston-area roads were blocked by high water. The city's two main airports were shut as the floods turned runways into ponds and more than a quarter million customers were without power as of Tuesday morning.
The Gulf of Mexico is home to half of US refining capacity. The reduction in supply led gasoline futures to hit their highest level in two years this week as Harvey knocked out about 13 percent of total US refining capacity, based on company reports and Reuters estimates.
The floods could destroy as much as $20 billion in insured property, making the storm one of the costliest in history for US insurers, Wall Street analysts say.
The Brazos River, one of the longest in the country, was forecast to crest at record highs well above flood levels on Tuesday about 30 miles (49 km) southwest of Houston, prompting authorities in Fort Bend County to order the evacuation of about 50,000 people.