Nice idea. You'll have to be careful buying. Fleece: "the coat of wool covering a wool-bearing animal (such as a sheep)." There's a lot of "fiber fill" out there these days. Usually made from some water repelling strands. None that I've seen (wife makes pillows/quilts) will soak up water the way you want. Quilts used to come with wool fleece, when people used them to stay warm at night instead of hanging them on a wall.
Looks like "fleece fabric" will get you sources of what you want. Not sure what country you are in, so I can't begin to make a recommendation. Check the descriptions carefully to ensure that you are ordering washed, felted wool, or at least untreated cotton. It might be best to give the company a call before ordering, as there lots of hydrophobic synthetics sold for "fleece". That is not what you want. The stuff has got to get soggy (hydrophillic).
For durability, wool is a better choice than cotton.