The trembler at 9.05am local time on Tuesday had an epicentre depth of 10km, the report added.
Crumbled shop fronts could be seen in a video showing the aftermath from the nearby town of Lhatse, with debris spilling out onto the road.
Reuters was able to confirm the location from nearby buildings, windows, road layout, and signage that match satellite and street view imagery. The date could not be verified independently.
Tremors were felt in Nepal's capital Kathmandu some 400km away, where residents ran from their houses.
Tremors were felt in the northern Indian state of Bihar which borders Nepal. As walls shook, people rushed out of their homes and apartments to open areas.
So far, no reports of any damage or loss to property have been received, officials in India said.
A magnitude 6.8 quake is considered strong and is capable of causing severe damage.
Southwestern parts of China are frequently hit by earthquakes. A huge quake in Sichuan province in 2008 killed almost 70,000 people.
According to China's state broadcaster CCTV, there have been 29 earthquakes with magnitudes of 3 or higher within 200 km of the Shigatse quake in the past five years, all of which were smaller than the one that struck on Tuesday morning.
In 2015, a magnitude 7.8 tremor struck near Kathmandu in neighbouring Nepal, killing about 9000 people and injuring thousands in that nation's worst earthquake.
