I certainly would test a piece of the fabric before using it. Put a drop or two of liquid soap (Dawn dish soap or Tide for clothes) in some very hot water and swish around the piece of red fabric - do this in a white bowl/container and you should be able to tell if it bleeds.
Many years ago, I made a black, white and red quilt for a friend - whose bedroom was those colors. Nobody knew to test the fabric first. Imagine my shock when she showed me the quilt a few months later and the white had turned pink due to the red fabric bleeding.