A government official, speaking on condition they not be named, said Baird spoke about blogger Raif Badawi's case directly with Prince Turki Al Faisal, a member of the House of Saud.
The high-level intervention came as Saudi authorities postponed Badawi's weekly flogging for medical reasons, a spokeswoman for the family said Thursday.
Family spokeswoman Elham Manea said that local doctors in Jiddah determined Badawi could not withstand the 50 lashes scheduled for Friday.
She added that Badawi's punishment won't likely go ahead for several weeks.
"The authorities conducted a very thorough medical examination and they came to the conclusion that he is not fit at all to be flogged, not this Friday, and it seems also not the following Fridays," Manea said in an interview from her home in Switzerland.
Badawi was sentenced to 1,000 lashes for criticizing Saudi clerics on a blog he founded, in addition to a 10-year prison sentence and more than $300,000 in fines.
The 32-year-old father of three received the first 50 lashes on Jan. 9, but his second set of lashes was postponed last week for medical reasons.
Amnesty International said the lashes were scheduled to be doled out over a period of 20 weeks.
Manea said Badawi's case is in the hands of the Saudi supreme court, which could order a re-trial. There is a chance, however, that the case will bounce between the courts several times before a final decision is made, she added.
"We are very cautious," Manea said. "We are very glad to hear this, but at the same time, we are very cautious. We are very worried that, nevertheless, things will change in Saudi Arabia."
Badawi is not a Canadian citizen but his wife and three children fled Saudi Arabia in 2012 and settled in Sherbrooke, Que., in 2013.
Baird raised Badawi's case during a bilateral meeting with the Saudi prince, one of several he had Wednesday at the World Economic Forum in Davos, Switzerland.
"While the Prince has no role in the government, he is part of a prominent think tank and holds a number of important positions in Saudi Arabia and internationally," the source said.
The prince is expected to visit Ottawa next month.
Meanwhile in Montreal, protesters called on the Canadian and Quebec governments to exert pressure on Saudi Arabia to intervene in the case.
About two dozen people, including federal and provincial politicians, attended a vigil in downtown Montreal Thursday afternoon, holding signs reading "Freedom for Raif Badawi" and "Lashes = Torture."
"What we ask (for) obviously is his freedom," said Beatrice Vaugrante, director general of Amnesty International Canada's francophone wing. "He's a prisoner of conscience for Amnesty International. He did not use any violence...he must be freed and reunited with his family here in Canada."
Quebec Premier Philippe Couillard, who spoke to Badawi's family last week, told reporters in Davos that he'd met with Baird regarding Badawi and they agreed their respective governments would work together to bring him to Quebec.
- With files from Mike Blanchfield in Ottawa and Julien Arsenault in Davos, Switzerland
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