The deal is worth $30 million, according to the NFL Network.
Whitworth, who turned 38 in December, is entering his 15th season in the NFL and is a four-time Pro Bowl selection and a two-time All-Pro.
After getting to Super Bowl LIII with arguably the NFL's best offensive line in 2018, Whitworth's play appeared to decline last season as the Rams' line went with more youth and experienced several injuries.
Whitworth committed 14 penalties, including 10 holds, amounting to the NFL's second-highest total and far more than the seven penalties he committed in the previous two seasons combined.
A key free-agent recruit in 2017 by Rams coach Sean McVay, Whitworth quickly asserted himself as a leader in the locker room and on the field. He also has provided guidance and feedback to McVay, who is four years younger than the veteran left tackle.
Before joining the Rams, Whitworth played 11 seasons for the Cincinnati Bengals, who selected him in the second round of the 2006 draft.
He also has distinguished himself in L.A. and Cincinnati because of his charitable work.
ESPN's Lindsey Thiry contributed to this report.