The highly-rated Fremantle assistant coach is understood to have addressed the Kangaroos board on Monday following an impressive presentation during the selection process.
Helping to guide minor premiers Fremantle to this year's grand final, it is the third senior coaching role the 43-year-old has put his hand up for after being pipped by Josh Fraser at Carlton and Mark McVeigh at Essendon.
The club has yet to confirm Graham's appointment, stating the selection process is still ongoing.
Media reports of Graham's looming appointment for the 2027 season surfaced after Collingwood great turned Geelong assistant Nathan Buckley pulled out of the coaching race on Sunday night.
Premiership coaches Adam Simpson and John Longmire did not pursue the role, with the latter leaving Sydney to become Essendon's general manager of performance under new Bombers coach McVeigh.
Other assistant coaches Daniel Giansiracusa (Hawthorn) and David Hale (Hawthorn) were reportedly in the frame alongside Graham.
North Melbourne's six-person panel is being led by football boss Todd Viney and North chief executive Jennifer Watt.
The appointment of a new senior coach was flagged by Viney at the club's awards night last week, with the club eager to end weeks of turbulence following Clarkson's axing.
Four-time premiership coach Clarkson, burdened with a record of just 20 wins from 82 games at North, was sacked in early September with a year left on his contract.
Viney said there were "gaping holes" in the team's game plan, while president Sonja Hood stated the next coach needed to be one that could "coach the modern game".
Clarkson accused North of dancing on his grave, but captain Nick Larkey backed the decision of the club's "bold, strong, courageous" leaders at their awards night.
Without a finals berth since 2016, North had started the season 8-7 but managed only one more win to slump to 14th.
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