"It is bittersweet," Finnegan told the newspaper. "For six years I played in the Titans organization. I am grateful and humbled that the Titans took a chance on me and the way they matured me over the years.
"But they wanted to move on. ... I don't know why, and I don't know what else a guy could do career-wise and in the community. But that's part of the business, and you have to move on. I'll have great memories of my time with the Titans. It just didn't work out."
Finnegan said he's not sure why the Titans won't make him an offer. The Tennessean reported that Finnegan's agent, Terry Watson, is seeking a contract similar to the five-year, $48.5 million deal Johnathan Joseph signed with the Houston Texans last offseason.
"The Titans told us they were not going to offer us a deal this year, so that is pretty black and white to me. They didn't franchise me. So this is all on the Titans; there is nothing that we could decline and say no to. It just baffles me that we never got anything from them. I don't know what I did to turn them off on me. I felt I did everything I could do," he told the newspaper.
While the Titans are ready to move on minus Finnegan, the cornerback's agent expects his client to sign a big contract with another team.
"I think Cortland could be one of the hottest free agents out there on the market, and I am eager to talk to teams once free agency starts," Watson told ESPN's Josina Anderson on Monday.
Finnegan, 28, has 14 career interceptions, with one coming last season when he didn't miss a snap. Finnegan played in the slot last season while Jason McCourty and Alterraun Verner played on the outside for Tennessee.
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.
He is known for his nasty on-field temperament and was fined by the NFL a few times in 2010, the highlight coming Nov. 28 in a fight with Texans receiver Andre Johnson when the cornerback took most of the punches and both men were fined $25,000.
Information from ESPN's Josina Anderson and The Associated Press was used in this report.