The Anfield favourite made just 12 starts for the Reds last season and has not been offered a new deal by manager Jurgen Klopp, although he has remained in training with the side while his future is decided.
According to The Sun, the 30-year-old is being eyed by Lazio - fifth in the Serie A last season - in order to strengthen their midfield, with his "experience and versatility" said to have piqued the interest of manager Simone Inzaghi.
The newspaper says that a potential move to the Italian capital has been made even more attractive due to the fact that Leiva already holds an Italian passport.
Having moved to Anfield from Gremio in 2007, Leiva is currently Liverpool's longest-serving player and has registered almost 350 competitive appearances for the club.
In a recent interview with the Reds' website, Leiva admitted that each successive pre-season with the club "gets harder and harder".