Hodgson met Cole and his representatives in an effort to thrash out the £90,000-a-week contract for the next three years.
Cole could also expect a significant signing-on fee if he moves to Anfield.
But with Spurs also in the hunt for the England midfielder, the 28-year-old is still considering his options ahead of the most high-profile Bosman deal of the summer.
Liverpool are desperate to secure a big name to prove to skipper Steven Gerrard and striker Fernando Torres they'll be competing for honours again this season.
Cole's preference appears to be to remain near London, but Hodgson hopes the prospect of becoming one of his most senior players will lure him north.
Liverpool's offer isn't a massive increase on what Cole was earning at Stamford Bridge, but the Merseysiders believe footballing rather than financial concerns will be the main factor in his decision.
The skilful midfielder spoke after the World Cup about regular first team football being his main priority, and the Merseysiders' can offer greater guarantees on this score than Spurs and Arsenal, who have greater strength in depth.
Liverpool hope Cole will make a decision by the end of this week as they welcome back their England contingent to training.
Hodgson would want the midfielder to immediately start preparing for the new season alongside Gerrard, Jamie Carragher and Glen Johnson who end their holidays on Tuesday
Joe Cole returned from holiday last night with the taunts of Carlo Ancelotti ringing in his ears that he had quit Chelsea for financial reasons.
The England international is expected to make a decision on his future in the next 48 hours, with Liverpool favourites to secure his signature.
Bookmakers yesterday suspended betting on Cole’s next destination as Anfield boss Roy Hodgson confirmed he had held talks with the playmaker, though sources close to the player have denied any deal has been struck.
There is no doubt Cole’s arrival on Merseyside would give Liverpool a huge boost. Hodgson confirmed initial talks had taken place, saying: ‘I’m not a betting man so it doesn’t even register with me that betting has been suspended.
‘I know we have been connected with him and it is good to know that we are connected with players of that quality but I don’t have anything further to report on the subject.
‘I haven’t had any discussions with him. Managing director Christian Purslow is the man looking after the transfers, not me.’
The offer by the Anfield club, though, is not the largest on the table. The £4.5million-a-year package is £700,000 a year short of that on offer at West Ham.
West Ham co-owner David Sullivan has always been reluctant to shell out huge sums on transfer fees, although he is less reticent about players’ salaries.
Spurs and Arsenal are also not out of the running and a switch to either would enable Cole to maintain an interest in Champions League football and not cause any major domestic upheaval.
But the Hammers now look a distant fourth in the running and the midfielder seems set to make a mockery of Ancelotti’s barbs and rebuff the highest offer to advance his career.
Ancelotti also claimed that he is no rush to add Liverpool’s Fernando Torres to his squad, saying he has sufficient firepower at his disposal without having to break the bank to land the Spaniard.
Meanwhile, Liverpool midfielder Javier Mascherano has still not responded to a voicemail left by Hodgson last week, which adds to speculation that he might be leaving to join Inter Milan.
The England star has been offered £90,000 per week for three years - over £13million in total.
if its footballing reasons he shouldnt be leaving.
ok what, nowadays being realistic is part of the game.
Originally posted by dragg:if its footballing reasons he shouldnt be leaving.
it is always difficult to play for a manager who has lost faith in you, right?
Nothing wrong to ply his trade elsewhere and asking for better salary. But come to United can?
Originally posted by jgho83:Nothing wrong to ply his trade elsewhere and asking for better salary. But come to United can?
haha, it seems MU is now, looking @ the free agents market, more than ever.
ironically, MU took a big gamble on MOwen, now with OwenH, will they dare to take another gamble?
Originally posted by Poignant:ok what, nowadays being realistic is part of the game.
exactly
he joined liverpool.