Black spots on white laundry (by 2 users)

Dear Repairperson,

When I removed my white laundry from the washer it had small black spots on all over it.  I re-washed them 3 times with triple bleach, I soaked the items in bleach water for 3-4 hours, and after that didn't work, I soaked the items in salt. All to no avail.  I'm afraid to use my washer for fear the spots will appear on all the laundry.  My washer is about 10-14 years old.  Kennmore model #110.82874120 (10 amps/60 Hz/120 volts/stock# 28741).  Could it be the deterioration of the inside of the black rubber-like hose that brings the water in and out of the washer?  I'm baffled and frustrated.  Could you please help?