The offensive material was also posted through doors of houses and taped to the centre in Longsight, south Manchester, the local library and St Luke's Primary School in nearby Ardwick.
Police believe the posters were distributed on Thursday and are treating the matter as a hate crime, the Evening Standard reported.
However, police did not specify what was on the posters or the nature of racism.
Police has released a CCTV image of a young man they want to speak to in connection with the incidents.
"Hate crime has no place in our community, and I would ask that anyone with information contacts the police.
"I am keen to identify a young man who was seen near to St Luke's Primary School on the Coverdale Estate in Ardwick around 12.45pm on Thursday February 16," he said.
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