ESPN's Dianna Russini first reported Cousins would start in Week 1 against the Miami Dolphins.
Cousins, 27, was drafted by the Redskins in the fourth round in 2012, and has played in 14 games with nine starts across three seasons in Washington. He started Washington's 31–13 victory over the Baltimore Ravens on Saturday, and in two preseason appearances has completed 20 of 26 passes for 245 yards with a touchdown.
Griffin was originally cleared to play against the Ravens, but after re-evaluation the team said he would need to sit out for at least another week. He worked out before Saturday's game and has practiced with the team this week, reports ESPN.
Sunday, an ESPN report claimed Redskins coaches and front office officials were ready to move on from Griffin and even explored trading him. The report said a Griffin trade was met with pushback from Washington's ownership.
The Redskins have denied the report.
Griffin, who was chosen second overall the same year Washington drafted Cousins, has completed 63.9% of his passes with 40 touchdowns and 23 interceptions in 37 games. Cousins has a 59.0 completion percentage in his career with 18 touchdowns and 19 interceptions.
The Redskins play their final preseason game on Thursday against the Jacksonville Jaguars.
- Erin Flynn and Rohan Nadkarni
GALLERY: RG III's House of Pain
RG III's House of Pain
As the saying goes, No pain, no gain, but Robert Griffin III seems to have suffered more than his share of the former with little of the latter since entering the NFL in 2012. Here, then, if you're brave enough, is a look inside the Washington quarterback's house of pain.
Pablo Martinez Monsivais/AP