The bench, which was installed at Arbournthorne Social Centre on East Bank Road in 2018, was set alight on Saturday, March 6 at around 5.40pm.
It was dedicated to Kelly Brewster, who died along with 21 others in the Manchester Arena terrorist attack on May 22, 2017, near to where she lived at the time of the tragedy.
After officers were alerted to the incident by the National Police Air Service, the fire then spread to part of the roof of the building, causing extensive damage.
Officers are continuing with enquiries to establish the circumstances surrounding the fire and identify those responsible.
PC Leigh Jenkinson from the Sheffield South East Neighbourhood team said: “We are committed to finding those responsible for this mindless act of vandalism, where a bench that had been dedicated to one of the victims of the Manchester Arena attack was destroyed.
"The fire also caused extensive damage to the Arbourthorne centre, which members of the community regularly use.
“At this time we do not believe this to be a deliberate targeted act of damage to the bench, but nevertheless this has been extremely upsetting for the family and we want to ensure those responsible are found.
"We are keen to hear from anybody who may have information that could help the investigation to please get in touch.”
If you are able to help please call 101 quoting incident number 716 of 6 March 2021, or call Crimestoppers anonymously on 0800 555 111.