Burress was released by the Giants on April 3, just over four months after the Super Bowl hero accidentally shot himself in the thigh with an unlicensed gun in a New York City nightclub.
Burress faces at least 3½ years in prison if convicted on a felony weapons charge.
Jets general manager Mike Tannenbaum said late Sunday night that "consistent with our normal approach of performing our due diligence, we contacted Drew Rosenhaus several days ago, prior to the draft, to inquire about Plaxico's pending legal situation."
Tannenbaum added that the Jets haven't spoken with Rosenhaus about Burress since.
Rosenhaus didn't immediately respond to a voice mail and e-mail left for him by The Associated Press seeking comment.
ESPN senior NFL analyst Chris Mortensen first reported Sunday that the Jets had internal talks about the possibility of signing Burress.
One of New York's biggest holes entering the draft was at wide receiver, where Jerricho Cotchery is the only proven veteran after the Jets allowed Laveranues Coles to become a free agent. They didn't address the position over the weekend, instead trading up Saturday to take quarterback Mark Sanchez with the fifth overall pick and again moving up to take running back Shonn Greene with the first selection of the third round Sunday. They took offensive lineman Matt Slauson in the sixth round with their only other pick.
Burress would give the Jets an experienced player with much-needed height -- he's 6-foot-5 -- at the position, but he certainly would come with lots of baggage. Burress was suspended by the Giants for the final four weeks of last season for conduct detrimental to the team and fined after the shooting. He also faces possible NFL sanctions for violating its personal conduct policy.
Burress's next scheduled court date on gun charges is June 15. At his last appearance on March 31, the judge postponed the hearing to give Burress' attorney, Benjamin Brafman, and the district attorney's office time to work on a plea agreement. A source close to the case tells ESPN's Sal Paolantonio that there has been no progress made in those talks.
He has 505 career receptions for 7,845 yards and 55 touchdowns in nine seasons with the Giants and Pittsburgh Steelers.
Information from The Associated Press was used in this report.