Free agent Cole has joined the Championship Rams on a deal until the summer.
Cole, who won 107 England caps and 13 major trophies, was a free agent after leaving LA Galaxy in November.
He said: “I had a conversation with Frank one-two months ago.
“He said: 'would I be interested in coming here?'.
“I have kept in touch with Lamps and seen how good the team has been playing and followed Derby.
“For me it was a no-brainer to come here and hopefully push for promotion.
“But it is going to be a challenge.
“I have not played in England for five years and it is a good tough league with strong players."