Both Jackson and the Rams explored trade possibilities but nothing came together before Tuesday's NFL trade deadline.
Jackson, 34, was willing to stay with the Rams but the team will give him a chance for more playing time elsewhere, the sources said.
Jackson's $2.75 million salary is guaranteed, so if he clears waivers the Rams would be on the hook for most of his contract, but that is subject to offsets on a deal with a new team.
The Rams have not announced the move, but Jackson indicated he and the team were parting ways in an Instagram post, writing: "I am very grateful for the time that I spent playing for the LA Rams. Although this isn't the outcome that we wanted, I am extremely appreciative for the opportunity & support given to to me by the Rams Coaches & all of the Fans. I understand that everything happens for a reason & I look forward to beginning the next chapter of my NFL career."
In need of a deep threat receiver, the Rams signed Jackson to a one-year deal over the offseason. Jackson's decision to sign with the Rams reunited him with coach Sean McVay, who was among Jackson's coaches with the Washington Football Team, and it served as a homecoming for the Los Angeles native who expressed an eagerness to win a Super Bowl in his hometown.
But the 14-year pro and three-time Pro Bowl selection has been used sparingly in the offense through the Rams' 7-1 start, catching eight passes for 221 yards and a touchdown.
ESPN's Lindsey Thiry contributed to this report.