Rahaf al-Qunun has said she fears her family will kill her if she was forced to return home.
Immigration police chief Surachate Hakparn said Wednesday evening that Alqunun's father and brother plan to remain in Thailand until they know what country she will be sent to.
Authorities in the kingdom insist it is a family matter although the embassy has been giving the case "care and attention" and it has been in contact with Ms Alqunun's father, a regional government official in the kingdom, "to inform him on her situation".
Australian officials have strongly hinted that Qunun's request will be accepted.
The Saudi embassy in Bangkok has not publicly commented on Ms al-Qunun's case since it initially claimed on Monday that she had tried to enter Thailand without the right papers, a charge which she denied.
The social media tsunami appears to have dramatically expedited her case.
Ms Alqunun's father and brother arrived in Bangkok on Tuesday evening and asked to see her but Thai officials said it was up to the UN's refugee agency.
"He wants to take his daughter back home, her mother fell sick after hearing her daughter ran away", Surachate said.
Ultra-conservative Saudi Arabia has some of the world's toughest restrictions on women, including a guardianship system that allows male family members to make decisions on behalf of female relatives.
At Bangkok's worldwide airport, security officials stopped her and confiscated her passport, which she said was later returned. She was referred to Australia for resettlement. "She didn't get the support from social media users, and that's why she's in Saudi Arabia now - she's disappeared", Alharbi said. She escaped from her family on a flight while they vacationed in Kuwait. She said access to twitter had "changed the game" in what was wished for her.
"My name is Rahaf Mohammed Mutlaq Alqunun, and this is my picture", she tweeted on Sunday from Bangkok.
A United Nations spokesperson told NPR that the refugee agency has had no contact with either family member but that the father and son are communicating with Thai authorities to try to meet with Alqunun. The 18-year-old captured the world's attention in a series of Twitter posts that at times read like an global thriller - with very real consequences.
"It has been reported that Rahaf had a Visa to go to Australia, and meant to apply for asylum".
In one of her first tweets from @rahaf84427714 she said she was in "real danger" if sent back to Saudi Arabia.
Under rules agreed at the United Nations 1951 Refugee Convention, member states are banned from sending someone with refugee status back to the country they are fleeing.
The latest incident comes against the backdrop of intense scrutiny on Saudi Arabia over the shocking murder of Saudi journalist Jamal Khashoggi a year ago, which has renewed criticism of the kingdom's rights record.
In an earlier and separate explanation released on Twitter, the embassy also denied sending officials to Suvarnabhumi airport to meet Qunun as she arrived via Kuwait or impounding her passport - as she alleged.
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