Forsberg would have to clear waivers for the Jets to send him to their taxi squad, according to The Athletic's Ken Wiebe.
The Hurricanes claimed Forsberg off waivers from the Edmonton Oilers on Tuesday. The Oilers signed him to a one-year, $700,000 contract in October before waiving him Monday.
Forsberg posted a .897 save percentage in three games with the Hurricanes last season. He spent the vast majority of the campaign in the AHL, going 15-9-2 with a .905 save percentage in 27 contests for the Charlotte Checkers.
The 28-year-old has also suited up for the Chicago Blackhawks and the Columbus Blue Jackets, who drafted him 188th overall in 2011.