Floods in the southern US city of Houston have forced thousands to flee their homes and overwhelmed rescuers who could not keep up with constant calls for help.

The incessant rain brought by Hurricane Harvey covered much of Houston on Monday.

Texas' largest city was submerged in visibly dirty water, with streets navigable only by boat.

In a frantic rescue effort, helicopters landed near flooded motorways, airboats crisscrossed neighbourhoods and high-water vehicles negotiated water-logged intersections late on Sunday.

READ MORE: Facing Hurricane Harvey

Some people used kayaks, inflatable beach toys and air mattresses to get to safety.

Volunteers joined emergency teams to pull people from their rooftops or from the water, which was high enough in places to gush into second floors.

Some formed human chains while others fired up their motorboats and jet skis to pluck their fellow Texans to safety.

The Harris county sheriff's office rescued more than 2,000 people in the greater Houston area using vehicles including motorboats, airboats and humvees on Sunday, a spokesman said.

Harvey made landfall late on Friday as a Category 4 hurricane and has continued to drop heavy rain as a tropical storm.

Parts of Texas are expected to see a year's worth of rainfall in the span of a week.

The storm has so far affected about a quarter of the Texas population, or 6.8 million people in 18 counties, Reuters news agency said, citing federal disaster declarations.

READ MORE: Harvey - Unprecedented flooding not over yet

The US National Weather Service said the "breadth and intensity of this rainfall is beyond anything experienced before".

Harvey was blamed for at least two deaths, and officials fear the death toll may rise as the storm triggers additional tidal surges and tornadoes.

Authorities ordered more than 50,000 people to leave parts of Fort Bend County, about 55km southwest of Houston, as the Brazos River was set to rise to a record high of 18 metres this week - more than 4 metres above its flood stage.

Robert Hebert, Brazos county judge, said the forecast crest represents a high not seen in at least 800 years.

"What we're seeing is the most devastating flood event in Houston's recorded history," said Steve Bowen, chief meteorologist at reinsurance firm Aon Benfield.

Greg Abbott, governor of Texas, said he will deploy an additional 1,000 National Guard troops for rescue missions. He had already called up 3,000 troops over the weekend.

Rescuers had to give top priority to life-and-death situations, leaving many affected families to fend for themselves.

One man was spotted clinging to a tree on Saturday night by police patrol officers who then formed a human chain across a bayou to save him.

Houston's George R Brown Convention Center was quickly opened as a shelter.

Most of the people at the shelter were African American or Latino.

Several people told the Associated Press news agency that in hindsight, they wished they had left Houston beforehand.

Before Harvey hit, local officials pushed back against Abbott's suggestion that people in Houston should leave and did not order a voluntary or mandatory evacuation.

Sylvester Turner, Houston's mayor, stood by that decision on Sunday, saying an evacuation would have put many people on roads that eventually flooded and endangered more lives than having residents stay in their homes.