Greens senator Sarah Hanson-Young, the first Australian MP to call for Australia to provide refuge to Ms al-Qunun, said it was "time to bring this courageous young woman to Australia to start her life as a free woman".
The UNHCR has since assessed her case and found she is a refugee.
He said her father and brother were scheduled to arrive in Thailand yesterday evening and that he would coordinate with the agency to arrange the meeting.
Thai immigration police chief Major General Surachate Hakparn said United Nations officials had expected her case would be concluded within days.
But armed with a phone, she barricaded herself into an airside hotel room and fought back - live-tweeting her fears of deportation in a campaign that swiftly galvanised global support and prompted a sharp U-turn by Thai officials. She also said she had renounced her religion, Islam.
While Human Rights Watch deputy Asia director Phil Robertson has backed this statement, Abdullah al-Shuaibi, charge d'affaires at the Saudi embassy in Bangkok, denied this. She set up the Twitter account after arriving in Thailand and being forcibly detained in a room at the transit hotel inside Suvarnabhumi airport.
"I will also advocate for the safe return to Australia of Mr Hakeem Alaraibi, who is now detained in Thailand".
After Alqunun left the airport, she was taken to a hotel in Bangkok where United Nations staffers were expected to interview her and process her claim.
"It's extremely concerning if it is the case that the visa has been canceled", she told The Associated Press, adding that Australia should allow Alqunun entry in any case.
The UNHCR issued a statement yesterday saying it would look into Ms Qunun's case "to assess her need for global protection" and that "It could take several days to process the case and determine next steps".
Ultra-conservative Saudi Arabia has come under fire since the murder of journalist Jamal Khashoggi in the country's consulate in Istanbul previous year.
Similar to Alqunun, Alaraibi was detained at Suvarnabhumi Airport while leaving after honeymooning in Thailand.
"I was told by UNHCR that there are several countries (offering asylum) but we can't be rushed", Maj.
The home affairs department said it will "consider this referral in the usual way, as it does with all UNHCR referrals".
Indications from Canberra suggest Alqunun may receive a sympathetic hearing. "Given the sensitivity of this issue, the process might still be prioritized and be processed quicker than usually".
Rahaf Mohammed Alqunun arrived in Bangkok on Saturday, January 5, appealing for asylum and her application was fast tracked for security reasons, partly because of the arrival of her father and brother.
The 18-year-old Saudi teen has been found to be a genuine refugee.
Even before the referral, Australian officials had been making positive noises about Alqunun's case. The strict legislation, however, only affects those attempting to enter outside legal channels. She was referred to Australia for resettlement.
While Saudi Crown Prince Mohammed bin Salman has been steadily easing restrictions on women, notably granting them the right to open their own businesses and to drive cars, the guardianship system remains in place and authorities remain sensitive to social codes.