Finnegan is scheduled to arrive in St. Louis on Tuesday, the sources said.
Jeff Fisher, who was hired as Rams coach in January, drafted Finnegan when Fisher was in Tennessee.
Finnegan, 28, has 14 career interceptions, with one coming last season.
The cornerback was a seventh-round pick out of Samford in 2006 who became a starter in 2007. The Titans extended his contract in August 2008, and he made more than $3.7 million last season.
Finnegan was fined by the NFL a few times in 2010, the highlight coming Nov. 28 in a fight with Houston receiver Andre Johnson when the cornerback took most of the punches and both men were fined $25,000.
The Rams released cornerback Ron Bartell earlier Tuesday.
Bartell was the latest casualty in the team's rebuilding efforts under new Fisher and new general manager Les Snead.
The 30-year-old Bartell was the team's top cornerback most of his seven seasons with the Rams, but missed virtually all of 2011 after sustaining two neck fractures in the opener. Bartell, who was injured when he absorbed a block by an Eagles lineman, has said he intends to resume his career.
Bartell was a second-round pick in 2005 and had been the second-longest tenured Ram behind running back Steven Jackson, a first-round pick in 2004.
Information from The Associated Press was used in this report.