Lalita Kayi, 50, was discovered a week ago in the dense forests of Sindhudurg district after her cries for help were heard by shepherds. They alerted the police who sawed off the chain and rescued her.
Ms Kayi, who appeared completely emaciated, was taken to hospital. Her physical health has since improved and, on Friday, she was moved to a psychiatric facility for further treatment, doctors treating her told the BBC.
In a written statement to the police, she has alleged that her husband "chained her and left her in the forest to die without food or water".
Police say they are looking for her husband in the southern state of Tamil Nadu on the basis of information she provided them.
But seven days after Ms Kayi was rescued, there is still no clarity on who she is, how she came to be in the forest, who tied her to the tree, and why.
Pandurang Gawkar, a cow herder who found her last Saturday, told BBC Marathi that he had taken his cattle to graze in the forest when he heard "a woman screaming loudly".
"The sound was coming from the forest on the side of the mountain. When I went there, I saw that one of her legs was tied to a tree. She was screaming like an animal. I called other villagers and the local police."
Police said that on her they found a copy of her passport, which stated that she was an American citizen, and her Aadhaar card - a unique ID for Indians - with her home address in Tamil Nadu.
They said she also had a mobile phone, a tablet and 31,000 rupees ($370; £290) in her possession - which allowed them to rule out theft as a motive.
Locals say that it was the woman's good fortune that the shepherd picked a spot near her to graze his flock that day. The forest she was discovered in is vast and she otherwise could have gone for days without anyone hearing her cries for help.
Police initially took her to a local hospital before moving her to a hospital in the neighbouring state of Goa.
Dr Shivanand Bandekar, dean of Goa Medical College, told The Indian Express newspaper that she had some wounds on her leg and that she appeared to be suffering from a mental health condition.
"We do not know for how long she did not eat, but her vital signs are stable," Dr Bandekar said.
On Friday, the woman's physical health had improved enough to be moved to a psychiatric hospital in Ratnagiri district of Maharashtra state.
"Currently, her health is stable," hospital superintendent Dr Sanghamitra Phule told BBC Marathi.
"She is taking medication, eating, and interacting with people. If she wants something, she can communicate it. She only knows English."
According to the police, Ms Kayi was a ballet dancer and yoga practitioner in America - some reports say specifically Massachusetts - and moved to India about 10 years ago to study yoga and meditation in Tamil Nadu.
It was there that she met her husband - in some media reports, police have called him Satish. Police say they believe at some point she fell out with her husband.
Some reports say that she stayed in a hotel in Goa for two days and then travelled to Mumbai city, India's financial capital.
But there is no clarity surrounding when or how she then ended up in the forest where she was discovered last week.
Ms Kayi, who was initially unable to speak, communicated with the police and doctors by scribbling notes on a pad. Through them she blamed her husband for tying her to the tree and claimed that she had gone without food and water for 40 days.
She also claimed that she had been given an "injection for extreme psychosis" which locked her jaw and prevented her from drinking water, and that she had to be provided nutrition intravenously.
"I am a victim and survived. But he ran away from here," she alleged.
Police say they have been unable to verify these claims and believe it is unlikely that someone would survive without food or water for so long.
They have registered a case of attempted murder against her husband and have dispatched teams to Tamil Nadu, Goa and Maharashtra to investigate the matter further. Her husband is yet to be traced by the police and hasn't made any statements to the media.
Police say they are also looking for clues in the mobile phone and the tablet they found on the woman.
The US embassy in Delhi - which media reports say has been "exerting pressure on the police to speed up the investigation" - has refused to comment on the case.
A spokesperson told the BBC that it could not respond to inquiries "due to the US Privacy Act", which governs the dissemination of private information.
Geeta Pandey & Cherylann Mollan