The Cessna, with an unresponsive pilot, later crashed in Virginia and officials say no survivors were found.
No survivors were found at the crash site on Sunday, Virginia state police said.
The Federal Aviation Administration said the Cessna Citation took off from Elizabethtown, Tennessee, earlier in the day and was headed for New York’s Long Island.
Inexplicably, the plane turned around over Long Island and flew a straight path down over the US capital before it crashed over mountainous terrain near Montebello, Virginia, at about 3:30pm local time (19:30 GMT).
Four people were onboard the Cessna, a source familiar with the matter told the Reuters news agency. A Cessna Citation can carry as many as 12 passengers.
It was not immediately clear why the plane was unresponsive or why it crashed.
The Cessna was registered to Encore Motors of Melbourne, Florida, according to the flight-tracking website Flight Aware.
John Rumpel, who runs the company, told The New York Times that his daughter, two-year-old granddaughter, her nanny and the pilot were onboard the plane. They were returning to their home in East Hampton, Long Island, after visiting his house in North Carolina, he said.
Rumpel, a pilot, told the newspaper he did not have much information from authorities but hoped his family did not suffer and suggested the plane could have lost pressurisation.
“It descended at 20,000 feet [6,000 metres] a minute, and nobody could survive a crash from that speed,” he said.
The US military scrambled F-16 fighter jets and attempted to contact the pilot, who was unresponsive, North American Aerospace Defense Command (NORAD) said in a statement.
The jets created a sonic boom over the US capital as they pursued the Cessna, officials said.
Residents of the city and its suburbs reported hearing the thundering noise, which rattled windows and shook walls for miles and caused social media to light up with people asking what had happened.
Several residents said they heard the noise as far away as northern Virginia and Maryland.
“The NORAD aircraft were authorised to travel at supersonic speeds and a sonic boom may have been heard by residents of the region,” the statement said, adding that fighter jets also used flares in an attempt to gain the pilot’s attention.
The US Capitol Complex in Washington, DC, meanwhile, “was briefly placed on an elevated alert until the airplane left the area”, Capitol Police said on Twitter.
The episode brought back memories of the 1999 crash of a Learjet that lost cabin pressure and flew aimlessly across the country with professional golfer Payne Stewart on board. The jet crashed in a South Dakota pasture and six people died.
In the case of Stewart’s flight, the plane lost cabin pressure, causing the occupants to lose consciousness because of oxygen deprivation.
Similarly, a small US private plane with an unresponsive pilot crashed off the east coast of Jamaica in 2014 after veering far off course and triggering a US security alert including a fighter jet escort.