Rapoport added the contract is worth up to $3.5 million.
The 25-year-old Fleming spent the first four years of his career with the New England Patriots after they selected him in the fourth round of the 2014 NFL draft out of Stanford.
He appeared in 47 regular-season games and made 20 starts during his tenure in New England.
He has played both tackle and guard in the NFL, and he started six games as a tackle last season as the Pats made it to their third Super Bowl in four years with Tom Brady leading the offense.
The two-time Super Bowl champion will likely start at right tackle for the Cowboys, according to Rapoport, which means La'el Collins could move to left guard.
Collins was Dallas' starting right tackle last season, but he appeared more comfortable and effective at guard in 2015 and 2016.
The Cowboys already have three Pro Bowlers on their line in left tackle Tyron Smith, center Travis Frederick and right guard Zack Martin.
Fleming will block for both 2016 Offensive Rookie of the Year Dak Prescott and 2016 rushing leader Ezekiel Elliott as Dallas hopes to return to the playoffs in 2018.