Rahaf Mohammed al-Qunun refused to board a flight from Bangkok to Kuwait on Monday and barricaded herself into her airport hotel room.
She said she feared her family would kill her for renouncing Islam.
She is now staying at a Thai government shelter while the UN refugee agency assesses her case.
Her father and brother have arrived but she is refusing to see them.
Renunciation of Islam, known as apostasy, is punishable by death in Saudi Arabia.
Thai immigration officials had initially said she should return to Kuwait.
In another development, Ms Mohammed al-Qunun said her passport had been returned. A Saudi diplomat had reportedly seized it when she flew into Thailand from Kuwait on Saturday.
She has told the BBC that she is fine, and hoping to be granted asylum in a third country.
She captured the attention of social media users around the world by live-tweeting every twist and turn in her story. Her brand new Twitter account attracted 50,000 followers in a day and a half.
"My life is in danger," Ms Mohammed al-Qunun told Reuters news agency. "My family threatens to kill me for the most trivial things."
Her father is the governor of al-Sulaimi, a town in the northern Saudi province of Hail.
A spokesperson for her family told the BBC that they did not wish to comment and all they cared about was the young woman's safety.
Human Rights Watch and other campaign groups have expressed grave concerns for Ms Mohammed al-Qunun.
She had travelled to Thailand for a connecting flight to Australia, where she originally hoped to seek asylum.
On Tuesday morning she retweeted her original appeal for asylum, adding in a subsequent tweet: "I want Canada to give me asylum."
The Australian government said it was "pleased" the UN refugee agency (UNHCR) was assessing her claim.
"Any application by Ms al-Qunun for a humanitarian visa will be carefully considered once the UNHCR process has concluded," a Department of Home Affairs official said.
The UN agency said it was "very grateful" that officials in Thailand had not deported her but her asylum claim would take "several days" to assess.
Saudi Arabia's embassy in Thailand has denied that its government is seeking her extradition, Reuters news agency reports.