Cole on Thursday underwent a medical at the club's training ground 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 [Thursday], top player, looking forward to it."
Watch: Joe Cole nears West Ham return... More Videos
West Ham have had a busy day in the transfer market as manager Sam Allardyce looks to strengthen his squad to ensure the Hammers do not slip towards the relegation zone following a solid first half of the season.
The Hammers, who are eight points clear of the drop zone in 11th position, signed Republic of Ireland Under-19 striker Sean Maguire from Waterford United on a two-and-a-half-year contract.
The 18-year-old, who was joint top scorer in the League of Ireland First Division in 2012, 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."
The Hammers also announced on Thursday that midfielder Yossi Benayoun has returned to his parent club Chelsea after making just six appearances for the club during a loan spell that was blighted by injury.
"West Ham United can confirm that the loan of @YossiBenayoun15 has ended and he has returned to @chelseafc," a statement on West Ham's Twitter feed read.
"The club would like to thank Yossi for his efforts during his second spell with the Hammers and wish him the best of luck for his future career."
West Ham co-owner David Gold revealed last week that Allardyce was keen to sign several players in the transfer window having seen his squad hit by a spate of injuries just before Christmas.
Reports this evening also claimed the Hammers had made a bid for Arsenal back-up striker Marouane Chamakh.