The Montreal Canadiens first spoke to the Jets about Kane in the off-season, and those discussions were revisited this season.
The New York Rangers, Philadelphia Flyers, New York Islanders and Florida Panthers have also kicked the tires on Kane's availability, along with a few other clubs.
It's believed the Canadiens have been among the more aggressive clubs, though the Flyers are supposedly right there with them.
The Flyers and Jets have been scouting each other for some time, prompting speculation that talks have been enhancing.
Flyers assistant GM Ron Hextall recently took in a Jets game in New York, while the Jets have a scout scheduled to watch tonight's game between the Habs and Flyers in Philly.
Kane, 22, is in the second-year of a six-year, $31.5 million contract. He is earning $4.5 million this season, but will make $6 million in each of the remaining four years of his deal, which does not include a no-trade or no-movement clause.
Cheveldayoff would move Kane for the right price, but that fee is believed to begin with two NHL-proven top-six forwards and a top prospect or a high draft pick.
The Flyers may be willing to part with the likes of center Brayden Schenn, right wing Matt Read, a prospect and a draft pick in exchange for Kane, while it's unclear what the Canadiens may be willing to move.
It's unclear how in-depth the Jets have gotten with other teams about Kane, but discussions are expected to carry out throughout the season, until either a deal is consummated or the Jets tell teams he's officially off the table.
In 26 games with the Jets so far this season, Kane has registered seven goals and seven assists for 14 points.