The epicentre of the quake was located in the Ionian Sea, near the Greek Island of Zante and about 50 miles north-east of the city of Pyrgos.
The quake had a depth of around 16 kilometres.
The US Geological Survey (USGS) said casualties are possible.
A witness said online: “Me and my partner are staying in Zante, it woke my partner up, and we’ve had three aftershocks much smaller, the larger quake has moved several items around the room.”
Another witness described the tremors to the European-Mediterranean Seismological Centre (EMSC) saying there was “extremely violent shaking”.
He added: “Objects fell off shelves.
According to reports, the quake lasted 15 to 30 seconds.
Local media reported that a 15th century monastery in the Strofades area of the island has been damaged.
Italy’s Il Messaggero news site reported that the earthquake was felt as far as southern Italy.
It claimed firefighters in the regions of Sicily, Calabria and Puglia received thousands of phone calls after the quake struck.
According to the USGS, a local small tsunami is possible and residents have been advised to move away from the shores.
A 6.8 earthquake is listed by the USGS as “very strong” with the potential for “moderate” damage.
The scientific agency claimed as many as 19.3 million people could have felt the earthquake, including 54,000 people who may have experienced “strong” to “very strong” shaking.
The USGS said: “Overall, the population in this region resides in structures that are a mix of vulnerable and earthquake resistant construction.
“Some damage is possible and the impact should be relatively localised.”
Greece lies in one of the world’s most earthquake-prone regions, and thousands of quakes are recorded every year.