Carr is in the final year of a five-year, $50 million deal he signed in 2012 as a free agent. He was set to count $13.8 million against the cap in 2016.
Carr's new base salary will be $4.25 million, the sources said. He will receive a $1 million roster bonus on May 1 and can earn up to $250,000 per game in roster bonuses. He can also earn an extra $500,000 in incentives.
The move creates roughly $3.6 million in salary cap space.
The move creates salary-cap room for the Cowboys, but it does not eliminate the possibility the team will draft Florida State's Jalen Ramsey with the fourth overall pick in Thursday's draft.
Carr has not had an interception since Thanksgiving 2013, but the Cowboys hold him in high regard. He has not missed a game in his career. He played through a right shoulder injury for most of last season.
With Carr's cut in pay, the Cowboys will have their top three cornerbacks returning.
Morris Claiborne signed a one-year deal as a free agent earlier in the offseason that could be worth as much as $3.75 million. Orlando Scandrick is returning from torn anterior cruciate and medial collateral ligaments that kept him out all of last season.
The Cowboys have shown heavy interest in Ramsey. They met with him at the scouting combine, put him through a private workout, and hosted him at Valley Ranch for a pre-draft visit.