Brian Quan's Toronto Star article "Canadian Air and Space Museum seeks a soft landing at Pearson" explains the current plans. The museum's biggest problems related to the lack of public knowledge of the museum's existence--I'd a friend who had worked at Downsview Park for an extended period but knew nothing about the museum--and relatively low attendance.
[T]he Greater Toronto Airports Authority agreed to help the struggling, and currently homeless, Canadian Air and Space Museum find a new site for its aircraft and artifact collection — including a replica of the legendary Canadian-built Avro Arrow fighter jet — at Pearson International Airport.
“We are talking and hoping we can come to a good arrangement,” said airport authority spokesman Scott Armstrong.
More than a year ago the aviation museum, a non-profit organization run by volunteers, was evicted from its Downsview Park location after running into financial problems and the park corporation’s desire to repurpose the building.
By September 2011, the museum owed the park’s property owner some $100,000 in unpaid rent.
“It was a very unpleasant experience,” Ian McDougall, museum chair, said of the eviction notice that forced volunteers to dismantle and pack the collection into storage containers.
The museum struck a deal with the GTAA earlier this year to temporarily store its collection on airport property.
With most of its collection already shipped there, McDougall said, the museum pursued negotiations to secure a permanent home at Pearson.
According to McDougall, the museum has already found a location at the airport where a new building could be constructed to house the collection. A press release notes the museum is looking at land near the south end of the airport, near Highway 401.
[. . .]
McDougall added that “we’re still working out large details,” particularly how the struggling museum can afford to lease airport land, given its financial difficulties. After a year with its doors closed to the public, the volunteer-run organization has all but burned through its small cash reserve, he said.
Earlier this month, in a move to turn dreams of an airport home into a reality, the museum launched a campaign on crowdfunding site IndieGoGo, seeking $500,000 in donations to help fund a new home for its collection, which includes a full-scale replica of the Avro Arrow fighter jet.
The crowdfundng page mentioned is here.