The Chiefs released Berry last week.
The five-time Pro Bowler, who suffered an Achilles injury in 2017, has only played four games over the last two seasons.
Berry has spent all nine years of his career with the Chiefs after they drafted him fifth overall in 2010. However, major injuries and a Hodgkin's lymphoma diagnosis in 2014 kept him sidelined for large chunks of time.
The Cowboys were rumored to be targeting Earl Thomas during free agency to fill their need at safety. Thomas, however, signed a four-year, $55-million deal last week with the Baltimore Ravens.