Baird made the comments after meeting with his Egyptian counterpart to discuss, among other things, the situation of the 40-year-old journalist who has spent more than a year behind bars in Cairo after he and two colleagues were arrested while working for satellite news broadcaster Al Jazeera English.
"I didn't leave Canada with any expectation that we would solve the issue today, but we had a good constructive dialogue and we look forward to continuing that and to resolving successfully this case," Baird told reporters at a news conference in Cairo.
"I think the (Egyptian) minister has an understanding of how important this is to me, how important this is to all Canadians."
Baird characterized his discussions with Egyptian foreign minister Sameh Shoukry as "fruitful," but noted that he did not anticipate an immediate end to Fahmy's ordeal.
"There's no two consular cases which are alike. This is a complex one," he said. "We are all working for seeing constructive resolution on that sooner rather than later."
The statements nonetheless came as a disappointment to Fahmy and his family, who had expressed hope that Baird's visit would mark the "finale" to what has been a nightmarish experience for the journalist.
"I understand that the ability of the Canadian government to help me is limited by the rules of diplomacy," Fahmy said in a statement released by his lawyers.
"But I do believe that Prime Minister (Stephen) Harper could do more to obtain my release if he were to directly intervene in our case."
Fahmy said his situation shows that anyone can fall victim to the political turbulence in the Middle East.
His fiancee also expressed disappointment.
"We expected that the deal would be sealed during Mr. Baird's visit," Marwa Omara told The Canadian Press. "However it seems that ... there is no decision made or nothing will be done during the visit."
Fahmy's family and Omara met Baird Thursday to express gratitude for his efforts, but the minister's stance still came as a blow.
"Mohamed didn't sleep since yesterday," Omara said. "He was putting high hopes on the visit...but we have to always keep managing our expectations."
The timing of Baird's visit was what led Fahmy and his family to believe a major development in the case was imminent.
A retrial was ordered this month for Fahmy and his colleagues, and Egypt's president had announced a new decree that gives him the power to deport foreigners convicted or accused of crimes.
Egyptian and Canadian government sources had also indicated that Fahmy's deportation process under the new law was in its "final stages," Omara said.
"We need the Canadian government to push for this," she said. "I don't know what the Egyptian and the Canadian government are waiting for. It's just journalists doing their jobs."
Baird noted later on Thursday that it would not have been in Fahmy's best interests for Canada to use "threats or tough talk" with Egypt on the matter.
He emphasized, however, that Fahmy would not be put on trial in Canada if he is deported from Egypt as a convicted criminal.
"That would not be an option that would be acceptable to Canada," Baird told reporters. "We want to see Mr. Fahmy return home as expeditiously as possible whether it is deportation, pardon ... (or) through a retrial."
Fahmy moved to Canada with his family in 1991, living in Montreal and Vancouver for years before eventually moving abroad for work, which included covering stories for the New York Times and CNN.
He took over as acting bureau chief of Al Jazeera's English-language channel in Cairo in September 2013 and, within days of taking the reins, raised concerns about the security status of the network's journalists working in the country.
He and two colleagues — Australian Peter Greste and Egyptian Baher Mohamed — were arrested on Dec. 29, 2013, and accused of supporting the banned Islamist Muslim Brotherhood group of ousted Egyptian president Mohammed Morsi. They were also charged with fabricating footage to undermine Egypt's national security.
The trio vehemently denied all the allegations against them but after a trial that was internationally denounced as a sham, Fahmy and Greste were sentenced to seven years in prison, while Mohamed was sentenced to 10 years.
Fahmy's family and a number of observers have suggested the case is largely a political one as Egyptian authorities have claimed Qatar-based Al Jazeera is biased towards the ousted Morsi and his Muslim Brotherhood — an allegation the broadcaster has rejected.
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