Cole underwent a medical at the club's training ground on Thursday after it emerged that Liverpool are prepared to cancel the 31-year-old's contract.
The Hammers, who brought Cole through their youth academy in the 1990s, hope the midfielder will sign his new contract with the east London club before midday so he can play in Saturday's FA Cup third round clash against Manchester United.
Should the deal go through, it will end a nightmare two-and-a-half-year spell on Merseyside for the former Chelsea player.
Cole has started just nine Barclays Premier League games since he swapped Chelsea for Liverpool in the summer of 2010, and spent last season on loan at French club Lille.
Reds boss Brendan Rodgers deemed the Londoner surplus to requirements when he took over last summer and Cole is keen to go back to Upton Park, where he spent four years in the youth system before going on to become a first team star for five seasons.
It is understood that Liverpool would pay Cole a negotiated settlement in order to get his estimated ?100,000-a-week salary off the wage bill.
West Ham midfielder Sebastian Lletget welcomed the club's move for Cole.
He tweeted: "Training with my man Joe Cole today, top player, looking forward to it."