I do not, as of yet, own an actual whistle, but having been a seamstress for over 3 decades, I feel somewhat qualified to attempt to add to the discussion. 
I would suggest trying a roll made of felted wool. This would be absorbent, protective, and breathable. If you wished to provide a bit more support, you could even add some pockets that would accept reinforcements (I am picturing something like chopsticks that would go between each pocket for your whistles-chopsticks becasue they are easy to obtain, durable, and replacable–you might want to sand the ends slightly so they are more smooth and rounded and slip into the pockets more easily, though).
To make it, you would want to obtain an old 100% wool sweater, in as large a size as you can manage (the process of felting shrinks that material significantly). I suggest a thrift shop for this.
Felt the material by simply tossing the sweater in the washing machine for a few cycles when you are washing something heavyweight (like your bluejeans or towels) and dry the sweater in your clothes dryer. Heat and friction are what do the trick. Your men’s large sweater will eventually shrink down to a youth size and it will have become thick and somewhat stiff. Before you attempt to use it you will need to 'block" it, this is simply laying the material out flat and pulling it into the shape you want (as square and flat as possible) and then letting it dry completely. If you find that it is trying to curl, you may need to pin it down to force it to stay flat, use stainless steel pins so you don’t get rust stains on anything.
Once it is dry, cut the material to the desired length and width. I suggest looking at jewelery rolls for ideas. Felted wool willnot ravel and the edges don’t really need to be finished, if you don’t want to do it.
I suggest a decent quality 100% cotton fabric about the weight that you would use for a summer shirt for the inside (if you want too make it really nice you could make bias tape with this fabric and use that to bind the edges and then make your ties from this). Prewash (I always prewash my 100% cotton fabrics in hot water and dry it and then do the whole thing again to make absolutely it is as preshrunk as it is going to go, nothing more frustrating than making a project and then some oaf washes and dries it and shrinks it all up!) and then iron with a bit of spray starch or sizing to help keep everything straight.
Place your whistles on the felted fabric and then place the cotton fabric on top, mark the spaces that you need to stitch the tubes for your whistles. If you want to add the additional protection of something like chopsticks that I mentioned earlier (you want them to be removable for ease in washing), you will need to also add these as well. I suggest using lots of straight pins and experimenting to find just the right size of pockets (make the pockets wider than it seems you will need, remove the whistles and then pin the two pieces of fabric together, add the whistles and then see how it performs, roll it up and see how it does and everything. Once you have the pockets at the perfect size and length, get ready to sew the pockets, just stitch down to make the pockets, if you want to ake it more durable, stitch a small upside down triangle at the top of each pocket, this will help to disburse the stress at the top of each of these seams, another option would be to stitch a ribbon along the outside of the wrap that goes all along the tops of each of these seams (thereby sandwiching the felt between the ribbon on the outside and the cotton fabric on the inside). Depending on how fat your final product is, stitch the middle of a length of ribbon either on one side or a few pockets in to make a tie to hold the whole thing together into a neat little bundle.
A few thoughts:
I would consider making the length of the roll maybe 2 inches longer than the longest whistle that you are designing the case for. You may want to also make a piece of cotton cloth that will act as a drape and will also add to the security of the case by stitching a piece of cotton cloth that is exactly the same size as the felt piece (finish the edges of this, I suggest by serging the edges and turning it under once so it does not become too bulky) to the TOP edge of the roll case, once your whistles are inside, flip this drape down and over the whole thing and then roll your case up-it will be impossible for whistles, especially conical ones, to work their way out. Particularly long cases may require two, three, or even more ribbon ties for security.
Using cotton and felted wool would defintely make this case breathable, but I would not store any musical instrument in such a case for long term storage.
I hope this makes sense, I am going by some craft projects that I have made for similar types of rolls, but for jewelery, artist brushes, and calligraphy pens. The directions above are basically exactly what I use for my calligraphy pens-most of which are far more exensive than the Pink Feadog that is going to be shipped to me after the first of the year
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Cheers!
Vic