The Newark Star-Ledger also reported on the situation Tuesday, adding that the team has made an offer.
Weatherford took part in several key plays in Sunday's game, including his first punt, a 36-yard kick that had the Patriots begin their drive on the 6-yard line. New England quarterback Tom Brady then was called for intentional grounding while standing in his own end zone, resulting in a safety and giving the Giants the first points of the game. Three of his four punts Sunday were downed inside the Patriots' 20-yard line.
The 29-year-old punter has played for five teams in his six seasons, most recently the New York Jets, who cut him after last season, with Jets special teams coach Mike Westhoff saying Weatherford "wasn't good enough." He's also spent time with the Jacksonville Jaguars, Kansas City Chiefs and New Orleans Saints.