I would use a free standing paper towel holder.

i have one of these, it's easy to move around by the bit that sticks out the top, it stands above mess, and i can put it away completely if i want. they come in different finishes.

http://www.bedbathandbeyond.com/1/1/19060-spectrum-euro-paper-towel-holder.html

If you have a drawer near by you can have it altered for this.

Oak Hollow Transitional Kitchen · More Info

If your under-cabinet lights are toward the front of your cabinets, you could probably attach a holder to the wall very close to the cabinet, someplace where you would be unlikely to plug something in.

Wall Mount Paper Towel Holder · More Info

Pennsylvania Industrial Paper Towel Holder · More Info

Stone County 901-484 Sassafras Black Paper Towel Holder · More Info

Thank you for the responses. I cannot replace a cabinet drawer and I do not have any space under the cabinets. I do not want to drill into the subway tile backsplash.

I will probably get a stand alone and keep it under the sink.

I agree; I would keep them in a holder inside the cabinets under the kitchen sink.

Went with a holder from Bed Bath and Beyond and placed it under the sink. It is actually working quite well!! I went with the Kamenstein Perfect Tear Paper Towel Holder (Stainless Steel). And if I decide to keep it on the counter top, it will go quite well!!