Typically, when you phone in to cancel a credit card, you will be switched to a “retention specialist” who will offer you something to not close the card.
This blog post has instructions for getting the best “retention offers” at Citi.
http://milestomemories.boardingarea.com/citi-thankyou-premier-retention-offers
Several months ago I collected all my Citi credit cards and phoned it to ask for retention offers. Every card had some sort of offer, and most were worthwhile. You can do this too.