The thirty-year-old has been in the cold at AC Milan, as Massimiliano Allegri has preferred to start with compatriot Robinho.
The Brazilian's lack of appearances on the pitch could lead him to seek a move for greener pastures as promises of a new deal have since been put on the backburner by the Serie A giants.
According to Itasportpress, Liverpool are looking at snapping him up, in an interesting tactical move to pair him up with Fernando Torres.
Purchasing the Brazilian would also fit John Henry's tight budget, as his contract will run out in June this year, making him a free agent.
However, his €7.5 million per annum wage demands could prove to be a stumbling block for Hodgson, which would leave Los Angeles Galaxy chief contenders to add him to their roster that includes David Beckham.