The 32-year-old has signed a 18-month deal with the Scottish giants.
Cole has been out of contract since last summer when he left the Hammers at the end of his contract.
“Being a Celt is not about the money, it’s about wearing the shirt with pride. It was the same at West Ham, if you put on that shirt and wore it with pride, the fans would take to you,” Cole told Celtic’s official website.
“I had a good rapport with the fans and that’s what I’m here for, as I can see the same sort of thing at Celtic.
“As soon as I came here everyone was explaining to me, even the taxi drivers, how much the club means to them, that’s what I wanted to hear and I want to hopefully be a hero to them.
“It’s a big relief, just be a part of such a big club and make friends, that’s what I’ve wanted since the summer started and when Celtic said they wanted me at the club, I just ran over.
“I’ve been up here alone for the past two or three weeks and I’ve settled in really well. The people of Glasgow are really welcoming and I’ve had just a great rapport with people.”
Cole came close to moving to the MLS during the summer.