First find out how far your town will allow the new fence to be away from your property line. If you have to be, say, one foot inside your boarder, you do not even have to have a talk with your neighbor because their chain link fence can stay put while you build your own fence. So, do check the town's fence regulations before you talk to your neighbor.
You can very well give your neighbor a choice. You can state that you are finishing up your fencing job and can either put the fence next to theirs or remove their fence, free of cost to place your new fence in the chain link fence location. Giving them a choice allows the to feel in more control. Although none of this is even necessary because you can build your own fence next to theirs without saying a word, as you do not need their consent or permission. Howsoever, you are very neighborly and kind to want to discuss this with them prior to moving forward. Good communication makes good neighbors but fences make GREAT neighbors.