The Italian international would not be available for Premier League duty until March due to a long-term knee ligament injury.
This has not deterred Brendan Rodgers, who believes that the ex-Manchester United front-man would link well with club talisman Luis Suarez.
Rossi, 25, is likely to available at a significantly reduced price due to his fitness issues and the fact that his club now ply their trade in the second-tier of Spanish football.
Although he would not be fully fit for some time, the Anfield hierarchy believe that the striker could make a significant difference in the closing stages of the season.
If a deal were to be pulled off, it would be quite a coup for the club, who would be securing one of the most talented forwards in Europe.
Prior to his injury, suffered 13 months ago, Rossi was thought of as one of the leading players in La Liga, attracting the attention of Barcelona as well as a host of top clubs in his homeland.
But, it’s unclear as to whether he will regain this form upon his return, or the likelihood of recurrences of the problem.
Although he represents Italy at national level, Rossi was actually born in the USA.
He began his European career with Parma, before switching to Manchester United in 2004.
The forward left the club three years later, signing for Villarreal in a £6.6m deal.