Your hand wraps will require washing periodically. Hand wraps that smell like a dead tennis shoe are offensive to their owner and to their owner’s training partners.
First: If they are not black or white they will bleed so don’t wash them with other colored clothes. If they are new soak them in solution of 80% water and 20% distilled vinegar for 1/2 an hour then air dry them prior to their first machine washing. That will set the color to reduce bleeding.
Second: YES do wash them in the washing machine. However do not just throw them in the washer…. they will come out completely tangled (yes… a rat’s nest!). Put them in a mesh bag (a lingerie bag) and wash them with like colors. DO NOT PUT THEM IN THE DRYER! Stretch them over the top of a door so they dry flat and let them air dry. Roll them up after they dry and you are good to go.