Officers want to hear from those who know is responsible for the graffiti which has appeared in Woodhouse Mill.
They are warning that the vandals may be forced to remove graffiti from other parts of the city as their punishment.
A South Yorkshire Police spokesman said: "-Do you recognise the tag? Do you know who may be responsible?
"The tag on an adjoining street reads the letters 'ANON', which may suggest the budding artist/s, may wish to remain anonymous. We therefore need your help in identifying them.
"No one wishes to become the victim of this type of crime let alone a victim who is elderly and suffering ill health.
"Paint has already been purchased to repaint the wall however due to the victim's current level of health they are unable to do this, causing upset.
"Considering suspects for removing graffiti from other areas is an option available to us if we are successful in identifying them"