Nibali is one of two active riders with titles on the Tour, the Giro and the Vuelta a Espana.
He ended the Giro, his home grand tour, in second place in May and the Sicilian had been waiting for the sixth stage finish at the top of the Planche des Belles Filles to assess his Tour de France form.
A winner there in 2014 -- the year he won the Tour -- Nibali struggled in the final part of the ascent, a brutal effort on gravelled roads in the Vosges region, losing 51 seconds to defending champion Geraint Thomas of Britain.
“He was still with the best with 500 metres to go but that’s when he got dropped,” coach Paolo Slongo told Reuters on Friday before the start of the seventh stage to Chalon sur Saone.
"It’s a strange situation because the GC (general classification) is not gone but he’s been distanced a bit."
Overall, Nibali lies in 20th place, 1:07 behind Thomas, the best placed of the main contenders.
“We will take it day by day and see how the sensations are,” Slongo, who has overseen all four of Nibali’s grand tour victories and another seven podium finishes, explained.
Attempting to win the Giro and the Tour in the same season is one of the most difficult challenges in the sport, with Italian Marco Pantani the last rider to achieve the double in 1998.
“The Giro was his first objective and then came the Tour. We knew that riding two grands tours would be difficult. He’s 1:07 behind Thomas, it’s not a disastrous situation."
Several contenders, including Colombian Egan Bernal, France’s Thibaut Pinot or Colombia’s Nairo Quintana, are ahead of Nibali.
Yet the 34-year-old has the ability to administer a killer blow and with six more mountain stages and four mountain-top finishes to come, there will be plenty of opportunities to turn the situation around.
“Letting go of the GC would be stupid,” said Slongo.
“We’ll see how things go but we will ride without stress, more free, and if the general classification is gone at some point we’ll see about winning stages.”
While Nibali’s performance was a bit disappointing, Slongo had plenty to celebrate on Thursday.
The Italian’s team mate Dylan Teuns won the stage from a breakaway and Giulio Ciccone, who finished second, claimed the overall leader’s yellow jersey.
“I was happy yesterday because we won the stage and also because a young Italian rider took the yellow jersey. It’s always a pleasure and a good thing for Italian cycling,” said Slongo, who has also coached Giro winner Ivan Basso, sprint specialist Elia Viviani and three-times world champion Peter Sagan, who are all Tour de France stage winners.