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.
The Hammers have completed the signing of Republic of Ireland Under-19 striker Sean Maguire from Waterford United on a two-and-a-half-year contract though.
The 18-year-old finished joint top scorer in the League of Ireland First Division in 2012, helping Waterford to gain promotion as runners-up.
Maguire, who scored 13 goals in 24 appearances, told West Ham TV: "In summer 2011 the Waterford manager brought me in for a couple of training sessions and gave me a chance in the first team and I took it.
"Here I am now at West Ham and it's a dream come true really.
"People tend to say my best trait is my movement so hopefully I can show that here. I haven't been told yet which squad I will be in but I presume I'll be going into the reserves so I'll try to work my way up like I did at Waterford."
Meanwhile, Chelsea have recalled Israeli midfielder Yossi Benayoun from his loan spell at West Ham.
"Yossi Benayoun has completed his loan spell at West Ham United and returns to Chelsea today," the club said in a statement on their website.
Benayoun, who last played for West Ham on November 11 due to knee trouble, has scored once for Chelsea in 11 appearances since he joined from Liverpool in 2010.