Jornet scaled Everest's north face, starting from the base camp on the Tibet side of the mountain, and reached the summit alone early yesterday.
"We believe Kilian established a new fastest known time for the route (from) Everest Base Camp," Laura Font from his press team told AFP.
"He climbed Everest without the use of O2 (or) fixed ropes and in one single push in 26 hours."
Jornet has previously set speed records on Mont Blanc, the Matterhorn, Denali, Aconcagua and Kilimanjaro.
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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 problems on his descent and stopped at a higher camp to recover.
"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 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 while the Spaniard started from the base camp 1,400 metres lower.
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