The new Liverpool boss sees the Chelsea reject as an ideal replacement for Yossi Benayoun, who has joined the champions for £5.5million.
Hodgson knows he has a battle on his hands to persuade the 28-year-old Londoner to leave the capital and is also concerned that Tottenham can offer Champions League football.
But he is ready to sell his energetic vision of the future for Liverpool, which includes trying to hang on to Fernando Torres and Steven Gerrard.
Cole has said he is looking for a new manager who "loves him" after falling out of favour with Stamford Bridge chief Carlo Ancelotti.
Spurs boss Harry Redknapp ticks that box having signed Cole as an 11-year-old at West Ham.
A Liverpool source said: "Roy knows he faces a tough job in persuading Joe to move north.
"But he sees him as the ideal playmaker to shape a new era at Anfield and hopes the player will be excited by this new challenge."
Hodgson is also ready to target Wigan's Honduran left-back Maynor Figueroa, 27, who would cost around £6million.
i wonder how far liverpool can go without money.
i dont doubt hodgson's ability but football nowadays is about money.
the best part about this deal is that he's free & is an english.
lfc shld better take him out
while selling off some fringe / under performing players
let's see if roy hodgson can do it!