Winston, 29, spent his first six seasons with the Houston Texans before signing with the Kansas City Chiefs last offseason. Winston was due to make $4.9 million in base salary next season.
He has started 103 games in his career including all 16 games last season for the Chiefs, which finished 2-14 and have the first pick in the NFL Draft next month. If Winston is signed, it could make Cowboys tackle Doug Free expendable. The Miami Dolphins also reportedly had interest in Winston.