According to Renaud Lavoie of RDS, Tortorella is one of the main candidates to replace Alain Vigneault, whom the Canucks fired last month.
Vigneault, meanwhile, is believed to be the frontrunner for the coaching job with the Rangers.
The Canucks have already spoken to former Dallas Stars head coach Glen Gulutzan, while former Tampa Bay Lightning coach Guy Boucher and former Buffalo Sabres coach Lindy Ruff are also believed to be considered for the job.
Canucks GM Mike Gillis also interviewed Dallas Eakins, but he opted to join Edmonton and the Oilers' new head coach over the weekend.
The Canucks may not have a new head coach in time for the June 30 NHL Draft, but it's believed Gillis wants to have a new bench boss in place by the start of free agency on July 5.
Tortorella has posted a 171-115-29 record in 315 regular-season games behind the Rangers' bench, since joining the club in the middle of the 2008-09 season.
The Rangers' 51 wins last season was a career-high for Tortorella. The team lost in to the New Jersey Devils in the Eastern Conference Finals.
The Dallas Stars are also courting Tortorella.