Montgomery's deal is worth a maximum of $12 million over four years and does not include a signing bonus, a league source told NFL Network insider Jason La Canfora on Tuesday.
The 29-year-old Montgomery started all 16 games last season, 14 at center and two at left guard. He joined the Redskins in December 2008.
The 30-year-old Scott has spent the last two seasons with Washington, although he was cut last November and re-signed a week later. He played in seven games as a backup in 2011.
The Redskins announced the moves Monday.
The Associated Press contributed to this report