Fox said late Friday the Rams will send their No 2 overall pick to the Redskins in exchange for Washington's No. 6 overall selection, future first-round picks and a second-rounder.
The trade cannot be announced or officially consummated until Tuesday under NFL rules, Fox said.
Washington has been in the market for a new quarterback and had been among the suitors for free agent Peyton Manning. The reported trade puts them in position to draft Griffin, the 2011 Heisman Trophy winner, as a long-term solution.
The Rams will stick with Sam Bradford under center and use their additional selections to fill weaknesses at other positions in the April 26 draft.
The Indianapolis Colts are expected to take Stanford quarterback Andrew Luck with the No. 1 overall pick. Minnesota drafts third.