Cousins believes he is ready to start and already has had discussions with the Redskins' hierarchy, which declared Griffin as their starter last month.
Cousins will not be eligible to be traded until March 11, but the Redskins are expected to receive calls from quarterback-needy teams.
One destination could be the Cleveland Browns, who plan to hire Kyle Shanahan as their new offensive coordinator, sources told ESPN on Saturday.
Shanahan has spent the last four seasons as Washington's offensive coordinator and worked with Cousins over the last two years.
Although he loves Washington and the organization, it is more important to Cousins to play, which is why he would welcome a trade to a team on which he would have that chance to start.
Shanahan and his father, former Washington head coach Mike Shanahan, were fired by the Redskins following a 3-13 season. Washington hired Jay Gruden as its new head coach last month.
Cousins has started four games and has appeared in eight contests overall during his two-year career, passing for 1,320 yards and eight touchdowns with 10 interceptions.