The spacecraft, known as Origins, Spectral Interpretation, Resource Identification, Security-Regolith Explorer (or commonly, OSIRIS-REx), has even snapped its first, blurry pic of the cosmic body, which is about the size of a small mountain, about 500 yards (460 meters) in diameter.
The spacecraft is designed to circle Bennu, and reach out with a robotic arm to "high-five" its surface, then return the sample it collects to Earth in 2023.
The first images of Bennu were taken on 17 August at a distance of 2.3 million kilometers from the $800-million-spacecraft.
"This is the closest we have even been to Bennu," said Dante Lauretta, OSIRIS-REx principal investigator at the University of Arizona, Tucson.
"This is significant in that we are now in the vicinity of the asteroid, closer than we have ever been even during the close approaches of the asteroid to the Earth."
Bennu was chosen from the some 500,000 asteroids in the solar system because it orbits close to Earth's path around the Sun, it is the right size for scientific study, and is one of the oldest asteroids known to NASA.
Astronomers say it poses a slight risk – a one in 2,700 chance – of colliding with Earth in 2135.
It is also a carbon-rich asteroid, the kind of cosmic body that may have delivered life-giving materials to Earth billions of years ago.
But it is the first asteroid-sample-return mission for NASA, and it aims to bring back the biggest sample ever, on the order of 60 grams.
The Americans who walked on the Moon during the Apollo era of the 1960s and 1970s collected and carried back to Earth 382 kilograms of moon rock.
In December 2018, the spacecraft will begin a detailed survey of asteroid's surface, which NASA has defined as its "arrival" at the asteroid.
Orbital insertion is expected on December 31. The sample, however, will not be taken until July 2020.