In the latest figures reported by the Xinhua newsagency, 32 people have been killed and 38 injured.
The quake hit at 9:05 am local time, with its epicentre located in Tingri, a rural county known as the northern gateway to the Everest region, at a depth of 10 km, according to the China Earthquake Networks Centre.
At least nine people had been killed on the Tibetan side, national television broadcaster CCTV reported earlier.
The impact of the tremor was felt across the region of 800,000 people, which is administered by Shigatse, the traditional seat of the Panchen Lama, one of the most important figures in Tibetan Buddhism.
Villages in Tingri reported strong shaking during the quake, which was followed by numerous aftershocks with magnitudes of up to 4.4.
Crumbled shop fronts could be seen in a video on social media 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.
A magnitude 6.8 quake is considered strong and is capable of causing severe damage.
Local government officials were liaising with nearby towns to gauge the impact of the quake and check for casualties, China's state-run Xinhua news agency reported.
Tremors were also felt in Nepal's capital Kathmandu some 400 km away, where residents ran from their houses.
"We felt a very strong earthquake. So far we have not received any report of injuries or physical loss," said Anoj Raj Ghimire, chief district officer of Solukhumbu district in Nepal, at the foot of Mount Everest, the world's highest mountain.
"We have mobilised police and other security forces as well as locals to collect information about the damage," he added.
The quake also jolted Thimphu, the capital of Bhutan, and in the northern Indian state of Bihar which borders Nepal.
So far, no reports of any damage or loss to property have been received, officials in India said.
Southwestern parts of China, Nepal and Northern India are frequently hit by earthquakes caused by the collision of the Indian and Eurasian tectonic plates.
A huge quake in China's Sichuan province in 2008 killed almost 70,000 people, while a magnitude 7.8 tremor struck near Kathmandu in 2015, killing about 9,000 people and injuring thousands in Nepal's worst ever earthquake.Â
