Lallana is one of the favourites of manager Jurgen Klopp, but with the competition for places so intense, especially in midfield, his chances are limited.
He has only played one game for the team this term, in the Carabao Cup.
Football Insider says the 31-year-old could be moved on this winter or in the summer.
The Reds can activate a clause to extend his deal, which expires this summer, by another year.
A winter sale is possible, but less likely as the club may not want to weaken boss Klopp's squad midseason.