The deal is worth $8 million and can reach as much as $12 million with incentives, a source told Schefter.
Webster, who visited with the Rams on Monday, will reunite with defensive coordinator Wade Phillips, his coordinator with the Denver Broncos the past two years, and may assume a starting role.
Webster, 26, celebrated his arrival to Los Angeles and thanked Phillips on Twitter.
Official a Ram ! Los Angeles what's good time to show up and show out ! Grind hard stay focused thankful— kwebb (@kayvonwebster) March 14, 2017
@sonofbum believed in me can't thank him enough for the opportunity ! Grind hard stay focused— kwebb (@kayvonwebster) March 14, 2017
Webster, a third-round pick by the Broncos in 2013, has recorded 64 total tackles in his career, but 59 of them came in his first two seasons. Over the past two years, he was stuck behind a dominant Broncos secondary that featured Aqib Talib, Chris Harris and Bradley Roby. The 5-foot-11 Webster has played only 58 defensive snaps since 2015, ninth among Broncos defensive backs during that span.
But the Rams, with roughly $15 million to spend on unrestricted free agents, are in need of depth at corner and are excited about Webster's athleticism.
Webster could potentially start opposite primary cornerback Trumaine Johnson, who is set to play a second straight season under the franchise tag but has also been shopped. The Rams' No. 2 corner last year, E.J. Gaines, has been limited to 10 starts over the past two seasons. Behind him are Lamarcus Joyner, who's better as a slot corner, followed by a former sixth-round pick in Blake Countess and two undrafted free agents, Troy Hill and Mike Jordan.
Webster has been very effective on special teams for the Broncos and was voted the unit's captain in 2016.
ESPN's Jeff Legwold contributed to this report.