"I'm not insulted," Horton told ESPN.com. "I was very happy Mrs. [Amy Adams] Strunk gave me an interview. I think she was very impressed with me. I was told some ideas I gave are going to be implemented. I'm still under contract. I'm talking about a contract extension. The insulted comment is not true. I'm very happy. I was very happy to be interviewed."
The team announced Saturday that it hired Mike Mularkey as its new head coach.
Horton, who has been connected to the defensive coordinator post in Cleveland, does not have an expiring contract, he said.
"Ray Horton is under contract to the Tennessee Titans," his agent, Brian Levy, told ESPN NFL Insider Adam Schefter. "Ray was extremely appreciative for the opportunity to interview for the head coach job. Comments about Ray's state of mind or feelings that are made by people other than Ray Horton are inaccurate and unfortunate and untrue."
The Tennessean quoted John Wooten, chairman of the Fritz Pollard Alliance, as saying Horton was insulted by the interview. The alliance promotes minority hiring in the NFL.
"The Browns had put in a request to interview him, and then Tennessee interviewed him [for the head-coaching job Saturday]," Wooten told the Tennessean. "And they said that they didn't know that he had any interest in being a head coach. Well, that's an insult to the man. That is just an insult.
"And then when he got in the interview, they seemed shocked that he was as prepared and knowledgeable and everything else at that point. And they of course already interviewed Mularkey on Friday, and they gave the job to Mularkey."
Horton was hired by former Titans coach Ken Whisenhunt in 2014. After a year as the defensive playcaller, his mentor, Dick Lebeau, was hired above him as assistant head coach/defense.
ESPN.com has confirmed Mike Mularkey will retain four members of the staff: LeBeau, linebacker coach Lou Spanos, assistant offensive line coach Mike Sullivan and assistant special teams coach Steve Hoffman. More, including Horton, could surface as Mularkey wades through his options.
Three assistants have moved on: Shawn Jefferson (receivers) to Miami, Giff Smith (defensive line) to San Diego and Nate Kaczor (special teams) to Tampa Bay.
Offensive coordinator Jason Michael has spoken with the Rams about a tight end or passing game coordinator position.