Esther Dingley, 37, had been walking solo in the Pyrenees near the Spanish and French border and was last seen on November 22.
The charity representing her family said on Friday it was “aware of the discovery” of the remains.
Following the reported uncovering of remains, Ms Bryant told the Sunday Times: “The dentist is sending a scan of Esther’s teeth. We have to send it to the consulate in Bordeaux.”
Ms Bryant had been due to return to the UK prior to the discovery but told the paper: “I’m now going to stay.”
Ms Dingley and her partner Dan Colegate had lived in Durham for years before they set off travelling around Europe in a camper van six years ago.
But before she went missing last year she had opted to hike in the Pyrenees alone while he stayed in a farmhouse in Gascony.
The pair last communicated on WhatsApp on November 22, when Ms Dingley sent him a selfie from the Pic de Sauvegarde.
Spanish authorities were reportedly alerted of her disappearance on November 25 and informed their French counter parts the next morning, having found her camper-van parked on the Spanish side of the border.
The news on the dentistry records comes after the Daily Mirror reported Spanish police sources had said “human bones” were found on Friday near where the Durham woman disappeared.
Speaking on whether they had also found any of her daughter’s belongings, Bryant said: “Nothing of her belongings was found.
“It’s upsetting that it’s not clear and definitive.”
But Mr Colegate, who is also in Bagnères-de-Luchon, refused to comment.
He said: “Until there is a confirmation either way, I won’t be making any statements.”
The Times also quoted French police chief Jean Marc Bordinaro as saying: “We cannot say anything at the moment because the discovery of the bones is too recent and they must be properly analysed”.