Jornet scaled Everest's north face, starting from a camp at 6,500 metres on the Tibet side of the mountain, in 17 hours late on Saturday, said a statement from the team.
"I'm so happy to have made the summit again! Today I felt good, although it was really windy so it was hard to move fast," he said in the statement.
"I think summiting Everest twice in one week without oxygen opens up a new realm of possibilities in alpinism and I'm really happy to have done it."
On Monday last week Jornet claimed a new speed record for an ascent without fixed ropes or supplemental oxygen when he summited the north face of Everest in 26 hours from base camp.
The 29-year-old had hoped to set a record for the fastest round trip - from base camp to the summit and back again - but was hit by stomach problems on his descent and stopped at a higher camp to recover.
AFP could not independently verify his claim. Climbs are usually only confirmed by the authorities once the mountaineer returns with proof such as summit photos and GPS data.
The China Tibet Mountaineering Association, which validates all Everest summits on the north side of the mountain, said it has yet to confirm Jornet's feat.
Climbing without using fixed ropes or oxygen is known as alpine-style mountaineering and is seen as a faster and lighter form of the sport.
Jornet has previously set speed records on Mont Blanc, the Matterhorn, Denali, Aconcagua and Kilimanjaro.
The Guinness World Records holder for the fastest ascent of Everest's north face is Italian climber Hans Kammerlander, who reached the summit in just 16 hours and 45 minutes in 1996 from the advanced base camp.
Hundreds of climbers have summited the 8,848-metre peak during the short spring climbing season which ends with the arrival of the monsoon in early June.
The first ascents of the season came unusually late in May -- delayed by high winds, fresh snowfall and extreme low temperatures.
Six climbers have perished on the mountain this year, including legendary Swiss climber Ueli Steck who fell from a ridge during an acclimatization climb in late April.