An easy and chemical free way to clean silver wire is to use aluminum foil, baking soda, salt, and hot water.

1.  Line a heat resistant pan or deep baking dish with aluminum foil, shiny side up.
2.  Boil a gallon or so of hot water and add a tablespoon each of salt and baking soda.
3.  Pour the mixture into your foil-lined pan and put your silver wire in, to soak.
4.  Make sure the silver wire is laying on or touching the aluminum foil for this to work.
5.  Remove the wire from the dish, run under tap water, and give a light buff with a paper towel.
While I’d like to say it took only seconds and that the wire came out ‘perfectly clean’… the truth is, depending on how tarnished your wire is, this might take up to 10 minutes of soaking time and for best results on some deep tarnish, it might take repeating the process after the toughest layers are removed.  Provided you have the patience to do this, it works quite well and best of all, cost next to nothing and doesn’t use a nasty chemical cleaner.
Here are some images for reference.  Enjoy and remember, you can also clean your silver jewelry this way, provided it doesn’t contain any natural stones that might be damaged by the heat or treatment.  This is also great for tarnished findings that are hard to polish by hand.