An official announcement of the move isn't expected until after the new league year starts next week.
Finnegan, though, tweeted earlier Wednesday a thank you to St. Louis fans, an apparent nod to the fact he had been notified of the move.
Finnegan, 30, signed a five-year, $50 million contract with the team in 2012. He was due a $3 million roster bonus on March 13.
By releasing Finnegan, the Rams will realize an instant savings of $4 million on the 2014 salary cap.
Should Finnegan sign a contract worth $2 million with another team, the Rams would reap that same number in cap savings, up to $3 million maximum. Finnegan's cap number for 2014 was scheduled to come in at $10 million.
According to the St. Louis Post-Dispatch, Rams coach Jeff Fisher said the team is interested in re-signing Finnegan at a reduced rate.
A seventh-round draft pick in 2007, Finnegan struggled mightily in 2013 before suffering a fractured orbital bone on Sept. 26 against the San Francisco 49ers. He attempted to play through it but ultimately landed on injured reserve after appearing in seven games.
Fisher and general manager Les Snead both indicated at the NFL scouting combine last month that Finnegan would be back in some form for 2014.
"He overcame a difficult situation," Fisher said. "I understand he's healthy now. And so we expect him back in the offseason program."
ESPN.com Rams reporter Nick Wagoner contributed to this report.