The Reds have been chasing the 23-year-old for months and reports are emerging claiming a £12.5m has been lodged for the Ivorian's signature.
The striker has played a key role in helping Lille to the top of Ligue 1 this season but the French club are bracing themselves for his departure in January.
Hodgson is desperate to bolster his side's attacking options as he feels they are too reliant on Ferrando Torres, who has suffered injury and form problems during the last year.
A number of names have been linked with a move to Anfield ahead of the transfer window reopening including Bolton star Johan Elmander and West Ham striker Carlton Cole.
Liverpool last week