The Redskins still must decide whether they will use their franchise tag or their transition tag on Cousins, a league source told Schefter.
Griffin would have to agree to a new contract with any team that trades for him, so the former Heisman Trophy winner figures to have some influence on where he would be traded.
Cousins was named the Redskins' starting quarterback over Griffin at the start of last season and led the Redskins to a 9-7 record and the NFC East division title.
Based on a projected salary cap of $153 million for the 2016 season, the franchise tag for the quarterback position would be worth $19.6 million while the transition tag would be worth $17.5 million. Teams have until Tuesday to apply the tags.
Redskins coach Jay Gruden and general manager Scot McCloughan both said Wednesday that they wanted to reach a long-term deal with Cousins.
But a team source told Schefter last week that the Redskins broke off negotiations with Cousins, and another source said that the sides were not close to an agreement.
Griffin was named Offensive Rookie of the Year in 2012, when he also led Washington to a division title. But he struggled with injuries and inconsistent play in 2013 and 2014 and did not appear in a game this past season.
Gruden also said Wednesday that he is disappointed that Griffin's stint with the Redskins did not work out better.
The Redskins picked up Griffin's $16.15 million option for 2016. If they are unable to trade Griffin, the Redskins are expected to release him by the time the new league year begins March 9.
The Redskins selected Cousins in the fourth round of the 2012 draft -- three rounds after taking Griffin with the second overall pick. But Cousins, 27, rose from being a backup in his first three seasons to a starter in 2015.
Before this past season, Cousins had started nine games and appeared in 14, throwing 18 touchdowns to 19 interceptions. His interception issues continued early in the 2015 season, as he threw six touchdowns and eight picks in the first six games.
But Cousins caught fire after that, finishing with 23 touchdowns and three interceptions in the final 10 regular-season games.
ESPN Redskins reporter John Keim contributed to this report.
Redskins will place tag on Cousins
Adam Schefter breaks down the Redskins' decision to place either the franchise or transition tag on quarterback Kirk Cousins and how they will move forward with Robert Griffin III.