Dallas was willing to give up a first-round pick, the source said, but the teams couldn't close the deal. Asked if the Jets wanted more than a first-round pick, the source told Werder: "Oh, yes."
Adams, who went to high school in Carrollton, Texas, would have been another boost to the Cowboys' defense, following the trade acquisitions of Robert Quinn and Michael Bennett, but he would have come at a much steeper price.
Adams, 24, was selected to his first Pro Bowl last season after finishing with an interception, 12 passes defensed, 3.5 sacks and 115 tackles. This season he has an interception, which he returned for a touchdown, five passes defensed and 39 tackles.
He has been outspoken about the Jets' losing ways since he joined the team as the sixth overall pick of the 2017 draft. The Jets are just 10-29 since he joined the team from LSU, where he was accustomed to winning.
The Baltimore Ravens also were actively going after Adams, a source told ESPN's Adam Schefter.
Adams is signed through 2020 and has a fifth-year option on his rookie deal for 2021.
NFL Network first reported the news of the talks between the Jets and Cowboys.