The Saudi embassy in Bangkok has said it did not demand the teenager's deportation and that the case was a family affair.
Thailand initially said it would deport her at the request of Saudi embassy officials, barring her from travelling on to Australia where Ms Qunun said she had meant to claim asylum.
But it may also have increased the risk to al-Qunun's safety - making her resettlement more urgent and strengthening her case for asylum.
Ms al-Qunun has also claimed her family would kill her if she were sent home to Saudi Arabia, where she has renounced Islam and "rebelled" against her father.
On Wednesday, Australia said it would consider taking Qunun in.
Thai immigration officials told Reuters that Canada had "granted her asylum", however Canadian officials told the BBC they now have "nothing to confirm" on the issue.
"We are providing necessary security for her", he said.
Ms Alqunun and her supporters drew global attention to her case through a social media campaign launched mostly on Twitter.
A UNHCR representative said "the process is still ongoing".
Payne said she also raised Australia's concerns with Thai officials about Hakeem Al-araibi, a 25-year-old former member of Bahrain's national soccer team, who was granted refugee status in Australia in 2017 after fleeing his homeland, where he said he was persecuted and tortured.
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On Sunday Ms Qunun told AFP her family was "abusive" and once locked her in a room for six months just for cutting her hair.
The Thai authorities' earlier willingness to return al-Qunun to Kuwait or Saudi Arabia could help her meet that standard. Under the kingdom's strict guardianship laws, adult women must obtain permission from a male guardian to travel overseas, marry or be released from prison.
Saudi woman Rahaf Mohammed al-Qunun before leaving the Suvarnabhumi Airport in Bangkok, Jan. 7, 2019.
"It would be a pretty stunning reversal if they didn't take her", he said.
Ms Alqunun deleted her Twitter account today, with a friend citing death threats for the decision. In a video clip of the meeting released by Thai immigration police, Alsheaiby is heard telling Thai officials: "From the moment she arrived, she opened a new account and her followers reached nearly 45,000 in a day. and I would have preferred it better if her phone was taken instead of her passport".
Rahaf Mohammed Alqunun's appeal for refugee status has underscored the severe restrictions that women in Saudi Arabia face.
"We are, as I've said, very concerned about his detention, very concerned about any potential for return of Mr Araibi to Bahrain", she said. He is obligated to do everything in his power to advocate, both privately and publicly, and to use the huge leverage that football has, with the Bahrain government, his own government, he's a Bahrainian national, and also with the Thai government to release Hakeem.
"[The] cause and reasoning [of both cases] are entirely different", she said.
"She needs to be protected from those who want to murder her, and should promptly be given asylum in Australia", the commentary said.
The Interpol red notice was withdrawn following Araibi's arrest based on his refugee status, but Thai authorities are still considering extraditing him following a request from Bahrain.