Think of the ropes of boats moored to a dock, but having to listen carefully (not loud and obvious) like 'creaking' or 'squeaking'.
It would have to denote friction without screeching - more like 'scratching' but muted.
'Rustle' implies some crackling absent in the tiny snapping sounds of fibers running past each other.
'Rasping' sounds human and clinical.
There are some far fetched options, such as 'susurring', which don't really work.
Example sentence: 'He doesn't feel like he is finally home for the night until he listens to the [click here for audio recording] sound of his shoelace knots coming undone while sitting at the edge of the living room sofa.'