Quintana and Movistar fell out, reported La Gazzetta dello Sport newspaper, and the Spanish WorldTour team offered to end its current deal that runs through 2019.
"No, it's not true," agent Giuseppe Acquadro told Cycling Weekly. "Yes, the other teams would want him, there's interest, but he has a contract through 2019 with Movistar."
The team, which helped Quintana to the 2014 Giro d'Italia win and to overthrow Chris Froome (Sky) for the 2016 Vuelta a España title, reportedly was upset with Quintana's desire to return to Colombia after the Giro d'Italia this year. He stayed in Monaco in June to train.
Quintana placed second overall to Tom Dumoulin (Sunweb) in the Giro. He lost the leader's pink jersey on the final day's time trial.
Movistar, said the article, also wanted Quintana to only race the Tour de France in 2017 and not aim for the double. This runs contrary to sources saying that it was Quintana who wanted only race the Tour and the team that pushed for the Giro/Tour double.
The 27-year-old was far from his best in the Tour, where he already placed second twice and once third to Froome. He recorded his worst result yet, 12th overall at 15-28 minutes, in Paris.
The team will race the Vuelta a España at home starting next weekend without a star leader. Alejandro Valverde is recovering from a broken leg suffered in stage one of the Tour and Quintana's Grand Tour season is over.
"Hence the breakup and the decision to put Quintana on the market," wrote the Italian sports daily. "This is a very expensive operation, because it is necessary to put €6 million on the table between the payment of the termination clause and new contract."
The article said that due to the rules, a decision would need to be made by Monday, August 15. And that team Astana, who are losing Fabio Aru, stand first in line for Quintana's signature.
Watch: Vuelta a España 2017 essential guide
"I'm not sure where that came from," Acquadro said. "I've not heard anything like this from Movistar.
"Quintana is important for Movistar the sponsor, which has a huge South American market.
"Also, Nairo was not at his best this time but you watch, he could still return to win the double one year. I think it's possible."
With Quintana staying put, Aru and Frenchman Warren Barguil make the biggest stage racer moves this transfer season.
Aru, who won a stage and wore the yellow jersey in the Tour this July, is due to sign with UAE Team Emirates and Barguil, victor of two stages and the mountains classification, is heading to team Fortuneo-Oscaro.
Froome extended his contract with Sky to 2020 before the Tour. Team-mate Geraint Thomas also added two more years.
Rigoberto Urán, second in the Tour, should announce his renewal with Cannondale-Drapac soon. And Romain Bardet (Ag2r La Mondiale), third, has an existing contract for 2018 and 2019.
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