Hodgson is working with under a strict budget at Anfield and most of the transfers his summer are likely to be outgoing ones, with Emiliano Insua the latest fringe player set to be moved on.
Hodgson has vowed to fend off any interest for Fernando Torres and Steven Gerrard, however, and although Javier Mascherano could move on, his aim is to strengthen his first XI.
Cole has already attracted interest from Arsenal and Tottenham this summer and the Reds may face a battle to lure him north, but the News of the World claims his representatives have met with Liverpool officials.
The Merseysides hope a significant signing on fee and the promise of guaranteed first-team football could persuade Cole to sign the three-year contract on the table.
It is also claimed Hodgson could make a move for Fulham's Paul Konchesky to replace the outgoing Insua.