I wouldn't take that risk
Some cotton rope may not fray,but the problem still remains that it isn't going to be digested and would cause a blockage far too easily.
I know a lot of people do rave about cotton rope for rats and birds,but it just isn't a good idea.You'd have to have rats that never even tested it with a nibble,and the nature of the beast makes it pretty much impossible.Some risks just aren't worth taking.