In a cellphone video posted to social media and presumably shot Sunday in the Dallas area, a fan spotted Adams in his Mercedes SUV and asked, "You coming to Dallas?"
Adams replied, "I'm trying, bro."
The Cowboys are among the seven teams Adams would welcome a trade to, ESPN's Adam Schefter has reported. Adams grew up in the Dallas area and attended Hebron High School in Carrollton, Texas, 31 miles from AT&T Stadium.
The Jets have not granted permission for a trade, so it would be a violation of anti-tampering rules if Adams were to contact another team.
The Cowboys emerged as suitors at the trade deadline in October, when they engaged in talks with the Jets. At the time, Adams told friends that he wanted a trade to happen, a source said. When it didn't, he claimed that the Jets betrayed him because he thought he was worthy of being an untouchable.
The Jets haven't commented on Adams' trade request, but sources said the team has no intention of trading him.
"It's time to move on," Adams said Friday on social media.
Adams, coming off an All-Pro season, is said to be frustrated by the lack of progress and communication in contract talks. He wants an extension that will make him among the highest-paid defensive players in the NFL. He has two years remaining on his contract, and the Jets don't appear to be interested in extending his deal until 2021.
The Cowboys could use a playmaker on defense, but they have only about $11 million in cap room. Quarterback Dak Prescott is counting $31.4 million, the amount of his exclusive franchise tag. If he signs a long-term deal, it could lower his cap charge and create more flexibility for the team to add Adams.
A source said Adams wants to play for the Cowboys so much that he might be willing to table extension talks.