The 24-year-old was offloaded to West Ham United at the start of the campaign on a season-long loan due to Rodgers being unable to fit the striker into his style of play.
While reports have suggested that Carroll will be on his way out of Anfield permanently this summer, it is understood that the frontman is willing to return to Merseyside.
"Andy is realistic about next season," a source told The Chronicle. "He is not expecting to be at Newcastle or West Ham.
"He's prepared to go back to Liverpool and fight for his place. Andy feels there is still some unfinished business at Liverpool and wants to continue his good rapport with the Kop."
While West Ham have been open about their desire to keep Carroll, manager Sam Allardyce has claimed that new financial fair play rules will prevent them from securing a deal.
Reports have also suggested that Carroll's former club Newcastle United are thought to be interested in re-signing the forward, but are unwilling to meet the Reds' £17m asking price.