New England has been looking to upgrade its pass rush and it believes it will do so with the 34-year-old Ellis, who has played 11 seasons with the Jets after New York picked him with the 12th overall pick of the 2000 draft.
The Jets had made a one-year offer for the veterans' minimum of $910,000, according to Ellis' agent, Mitch Frankel.
Ellis has earned respect throughout a career in which he has piled up 72½ sacks, ranking third on the Jets' career list behind Joe Klecko (77½) and Mark Gastineau (107½).
Ellis has been a reliable player, missing just six games since becoming a starter in 2001. His ability to play all along the line made him a valuable part of Rex Ryan's aggressive Jets defense.
The Jets used their first-round pick to select Ellis' heir apparent, Muhammad Wilkerson, who has been working with the starting unit in training camp. Ryan has praised Wilkerson, saying he expects the former Temple standout to be the opening-day starter. Ryan had said that he would have liked to bring back Ellis, although he made it clear that his playing time would be reduced.