Graham, a former Australian Football League star for Geelong Cats who was in his fourth season with the Jets, averaged just 27.3 net yards in the 19-10 loss. The Patriots benefited from excellent field position throughout the game, starting five of their nine offensive series in Jets territory.
Jets coach Eric Mangini, who discovered Graham in an AFL kicking competition in Australia 11 years ago, made it clear on Monday that he was disappointed with the punting situation.
"We'll look at it very closely, then we'll move forward here," he said, "see where the different opportunities are and evaluate them."
The 34-year-old Graham signed with the Jets in 2005 and won the job in training camp with his booming left leg and unorthodox end-over-end kicks. At 31, he also was the oldest rookie to make an opening day roster.
The Jets signed him to a six-year deal in 2006, and he was selected a special teams captain. He struggled last season with a career-low 43.3 average and 36.6 net average, and was also benched for one game.
Graham didn't help his cause in the season opener at Miami when New York was left without a place-kicker after Mike Nugent was injured. Many teams use their punters to kick field goals and extra points, but Graham was unable to do it capably because of his inexperience with the kicking motion and technique.
Graham appeared in 49 career games for the Jets, averaging 43.7 yards with 24 touchbacks and 67 inside the 20-yard line.