Jornet scaled Everest's north face, starting from the base camp on the Tibet side of the mountain, and reached the summit alone early.
"We believe Kilian established a new fastest known time for the route (from) Everest Base Camp," Laura Front from his press team told AFP.
"He climbed Everest without the use of O2 (or) fixed ropes and in one single push."
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.
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 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 problem on his descent and stopped at a higher camp to recover.
"Reaching the summit of Everest without fixed ropes isn't something you'd do every day! I saw a fantastic sunset and finally reached the summit at," Jornet said in a statement.
"Up to 7,700m I felt really good and was making progress as planned but then I started to feel unwell, probably from a stomach virus. From then on I made slow progress and had to keep stopping to recover."
Fellow climber Adrian Ballinger, who is at base camp with Jornet, told AFP the Spaniard might try again in the next few days to complete a recording-breaking round trip climb.
The China Tibet Mountaineering Association, which validates all Everest summits on the north side of the mountain, could not be reached to confirm Jornet's feat.
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.
But Jornet's team said Kammerlander had begun his ascent from Advanced Base Camp at 6,500 metres (21,325 feet) while the Spaniard started from the base camp 1,400 metres lower.
There are hundreds of climbers currently on Everest hoping to summit the 8,848-metre peak before the short spring climbing season 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.