It is understood Cole’s salary works out at around £15,000 a week including bonuses and he will complete the formalities of his move to Pride Park early next week.
Lampard has pushed for Cole’s signing as he wants defensive cover, with Derby in the play-off positions ahead of Saturday’s home game against Reading.
The Derby manager said: “He would bring quality and experience. I speak to Ash all the time. He’s a great lad and I know he’s fit, fresh and wanting to continue to play.
“I know the influence he would have but it has to be right for all parties. In my opinion, he’s the greatest left-back of a generation.
“Of course, his age is an obvious thing but it’s not a problem for me. I think it’s pretty clear that we’re interested in Ashley – but nothing is done yet.”
Cole started his career with Arsenal in 1999 and won two Premier League titles and three FA Cups, while he was also part of Arsene Wenger’s ‘Invincibles’ team in the 2003-04 campaign.
He completed a controversial move to Chelsea in 2006 and enjoyed further success, winning the Champions League and the FA Cup four times.
Cole, who won 107 caps for England, then moved to Roma before joining LA Galaxy in 2016.