Elliott will see his current contract expire in the summer, meaning he will be available as a free agent.
The 29-year-old has spent the last seven years at Ashton Gate after joining from Millwall in 2007.
Elliott has made over 400 career appearances after bursting onto the scene as a 19-year-old with Millwall and is a regular member of the Jamaica national team.
Elliott has been a mainstay in the City side and a number of sides higher up the league are thought to be keeping tabs on his situation.
"I am out of contract in the summer and I will wait and see what happens between now and then," Elliott told Sky Sports.
"I have loved my time at the club, especially the fans, but I feel at this stage of my career I need a new challenge and I am ready to start a new chapter.
"I am keeping my options open and we will just see what happens."
Elliott, who has played the majority of his career in the Championship prior to Bristol City's relegation to League 1, is determined to play at the highest level possible.
"I am excited about what the future holds for me and I have got aspirations to play as high as possible for as long as possible," added Elliott.