File a fraud alert with the credit reporting agencies and also get your credit card number changed. I think these people are credit card thieves. Since they can spoof your phone number, they can call your credit card provider and make it look like the call is coming from your phone. I think all they need is the last four digits of your credit card to access your account. That is one racket, asking for your card number is another. If you reported your card stolen, they should not be able to do anything with it and hopefully they will get caught when they try to buy something with it.
My new approach is to pretend interest and, when they ask for my credit card information, tell them I have to go find the credit card. Then I let them sit and wait while I work on the computer or do something else. In a couple of cases (after they've waited several minutes) I tell them I am on the Do Not Call list and never had any interest in their so-called service.
Same thing here. They called my cell phone, which is on the national do not call list. Automated message: "Cardmember services... no problem with your card, but we'd like to offer you a lower interest rate..." Pressing 1 got me a human, who I asked to put me on their "do not solicit list." Got an identical call from 803-692-1875 on 8/17/2011 at 1:30pm EDT.
Automated message to my cell phone offering a lower credit card interest rate -- the message then said to press 1 to speak to a live operator. The only credit card companies I use would not contact me in such a manner. My cell phone number is on a do not call list, so I am not certain how they got it.