The 26-year-old Sydney defender has nominated the Kangaroos as his preferred destination after also meeting with Geelong, St Kilda and Gold Coast.
North turned Blakey's head with a reported eight-year offer worth more than $11 million, before he toured Arden Street and completed a medical on Friday.
Sydney will still need to agree to move the 2024 All-Australian, who will likely command a hefty price at the trade table with five years remaining on his contract.
The Kangaroos hold picks No.5, 45, 64, 72 and 82 in this year's draft, plus future first-round picks in 2027 and 2028.
Swans list boss Chris Keane did not rule out listening to trade offers, but suggested brokering a deal would be "significantly challenging".
The dashing half-back's father John Blakey played 224 games for North Melbourne, including their 1996 and 1999 premierships.
Blakey was one of five Sydney players who served a club-imposed ban through the AFL finals for breaching the Swans' behavioural standards at the Pullman Hotel in East Melbourne.
The players' punishment was separate to an ongoing Victoria Police investigation into an alleged sexual assault at the hotel.
No charges have been laid, and AAP is not suggesting any wrongdoing.
North Melbourne have also been in talks for Nick Murray after the young Adelaide defender requested a trade to Arden Street.
Blakey's trade request came as the league approved a mega trade involving Port Adelaide, the Bulldogs, Melbourne, Hawthorn and Carlton, two players and scores of picks.
Under the five-team deal, the Bulldogs gave up picks No.12 and No.31 and a 2028 first-rounder to bring in Hawthorn's Finn Maginness, Carlton's Hudson O'Keeffe and later picks in this year's draft and future years.
Port parted with picks No.4, 46 and 81 and six second-round or worse future selections to get back picks No.6 and No.10, Melbourne's 2027 first-rounder and the Dogs' 2028 first-rounder.
The Demons surrendered picks No.6, 10 and 87 and their 2027 first-rounder - as well as two late future picks - for picks No.4, 12, 31 and 46.
Hawthorn (2027 third-rounder) and Carlton (2028 fourth-rounder) received late future picks for Maginness and O'Keeffe.
Separately, the Bulldogs lodged paperwork to acquire Butters as a free agent.
With the mega trade approved, Port are expected to not match Butters' contract offer and accept the subsequent first-round compensation pick.
The Power need draft capital to match Essendon's expected pick No.1 bid on Port academy prospect Dougie Cochrane and 2027 prospects Zemes Pilot and Louis Salopek.
Butters' arrival at the Bulldogs would be a significant boost to their 2027 premiership chances, with star forward Sam Darcy set to return from injury.
Elsewhere, Gold Coast should secure Hawthorn pair Aidan Schubert and Sam Butler, easing the Suns' call to accept a compensation pick (currently No.7) for Ben King.
Brisbane traded second-year Lion and 2025 premiership player Sam Marshall and pick No.39 to St Kilda for a 2027 first-rounder.
West Coast are still haggling with Melbourne over veteran Jack Viney, but any deal with the Demons for West Australian native Trent Rivers is likely to be separate.
The Swans have delisted veteran Harry Cunningham, with the 233-game defender to join Adelaide via the rookie draft on a two-year deal.
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