The Dallas Morning News reported the Cowboys made Durant their first free-agent signing of the year, reaching an agreement with him on a two-year deal.
The newspaper said the team has salary cap considerations that will mean Durant will have to wait to sign a contract.
Still, he posted a message on Twitter saying he would be having to "learn that South Dallas Swag."
The Detroit Free Press noted that while Durant had a career-high 103 tackles last year and played all 16 games for the first time in his career, the Lions instead go with DeAndre Levy, whom they re-signed, along with Ashlee Palmer and Tahir Whitehead.
Swim Week Miami Beach 2014 [PHOTOS]