Some high-ranking Redskins officials and coaches want to part ways with Griffin, who has been sidelined with concussion symptoms, but are meeting resistance from the team's ownership, team and league sources told ESPN on Sunday.
Gruden commended Griffin, who entered training camp and the preseason as Washington's starter, for his hard work but said Cousins has earned the starting job.
"I think all three quarterbacks should be commended for their efforts, their willingness to get better," said Gruden, referring to Cousins, Griffin and Colt McCoy. "But when it's all said and done ... we feel like at this time Kirk Cousins gives us the best chance to win."
"There's something powerful about feeling believed in, and something powerful about knowing where you stand," Cousins said.
Cousins also praised Griffin for the example and leadership he has provided.
"He remains classy, he works very, very hard and I learn from that example," Cousins said.
Gruden said Griffin, who missed Saturday's preseason game, is still in the NFL's concussion protocol. When Griffin is cleared to return, Gruden said the Redskins would determine whether the Heisman Trophy winner or McCoy would be the backup quarterback.
"We believe in Robert," Gruden said. "It's a good problem to have -- to have three quarterbacks that are competing and working their butts off."
Cousins has started nine regular-season games over his first three seasons in the NFL. The 27-year-old said that he realizes Monday's announcement does not mean he will necessarily keep the starting job throughout the season, especially if he performs poorly.
"It's a step in the process," Cousins said. "It doesn't feel like the finish line by any means."