Hainsey, 32, is scheduled to become an unrestricted free agent July 5.
The Sun claims he likely won't be brought back to Winnipeg as a result of the team's young blueline.
Hainsey is wrapping up a five-year, $22.5 million contract that had an annual salary cap hit of $4.5 million.
He could see his salary climb above $5 million this summer, as several teams are looking to bolster their defensive corps.
The Detroit Red Wings, Philadelphia Flyers and Carolina Hurricanes are among the teams interested in strengthening their blueline and could target Hainsey this summer when the free agency doors swing wide open.
Meanwhile, the Jets are expected to deal young winger Alexander Burmistrov around the NHL Draft.
Burmistrov has reportedly asked to be traded, and the Jets are weighing all options.