As public pressure heightened, an Australian minister appeared to go beyond Canberra's initial bureaucratic promise to consider her case if and when United Nations experts judge her fear of mistreatment justified.
The woman fled from her family and expressed fears for her own life.
The UN's refugee agency said on Tuesday it was investigating the case of the teenager.
In a Twitter update on Wednesday, Ms al-Qunun thanked her 107,000 followers for their "support in my hard psychological situation" and said that she had "regained my strength" after a dramatic few days.
"It would have been better if they had confiscated her mobile instead of her passport".More news: The Accidental Prime Minister: Anupam Kher on how he became Manmohan Singh
Australia national broadcaster ABC reported that the country's Home Affairs Department announced late Tuesday that it would consider Alqunun's application for asylum if she was found to be a genuine refugee, and called on the Thai authorities and UNHCR to assess her claim as quickly as possible.
The teen wrote of being in "real danger" if she is forced to return to her family, posting a copy of her passport to prove her identity.
But Rahaf Mohammed al-Qunun's father and brother would have to wait and see whether the United Nations refugee agency would allow them to see her, immigration chief Surachate Hakpan said.
"The government will be making no further comment on this matter".
"We have no idea what he is going to do", he said.
Ms Al-Qunun's Twitter account has attracted tens of thousands of followers in less than 48 hours and her story grabbed the attention of governments, activists and well-known figures all over the world.
Thailand is not a signatory to the UN Refugee Convention and provides no legal protection to asylum seekers, although there are more than 100,000 refugees in the country.More news: New trailer for The Punisher season two is a testosterone-fueled bloodbath
Thailand's immigration police chief, Major General Surachate Hakparn, told reporters today that Saudi diplomats told him they are satisfied with how her case had been handled.
Global pressure has mounted on Thai authorities to keep Alqunun safe and to ensure she isn't forcibly returned to the Saudi kingdom, which has been subject to worldwide condemnation over the killing of Washington Post writer Jamal Khashoggi at a Saudi consulate in Turkey.
"He wanted to make sure that his daughter was safe. he told me that he wanted to take her home", he said, adding that her father denied Ms Alqunun's allegation that her family was abusing her physically and emotionally.
Saudi Arabia enforces male guardianship laws, which require that women, regardless of age, have the consent of a male relative - usually a father or husband - to travel, obtain a passport or marry. So while on vacation in Kuwait, Alqunun slipped away from her family and tried to fly to Australia through Thailand. It said the embassy is not communicating with the teenager, but is communicating with Thai authorities.
On Sunday Qunun told AFP her family was "abusive" and once locked her in a room for six months just for cutting her hair.
In some cases, Saudi authorities were involved in forcing women to return to their families and in other cases, local authorities suspected the women of seeking asylum and deported them, the activist said. In the meantime, her al-Qunun has asked that the media and public continue to pressure officials to follow-through on securing her asylum.
Saudi Arabia's human rights record has come under intense scrutiny since the killing of Saudi writer Jamal Khashoggi in October.More news: Chelsea boss Sarri hits back at Bayern after Hudson-Odoi comments