The daytime attack at St Simon's Catholic Church left it with "significant damage" to its interiors.
A police spokeswoman said in a statement: "Between 1.30pm and 4pm on Monday, April 29, vandalism was carried out at St Simon's Church at Partick Bridge Street.
"The church would have been open at the time."
"There is nothing to suggest the incident is sectarian or hate crime. Inquiries are ongoing."
Disgusting destruction in St Simon's Catholic Church Glasgow. Catholics just want to worship in peace and we are seeing more graffitti and vandalism. What are the parties doing about these issues? @NicolaSturgeon @LabourRichard @RuthDavidsonMSP pic.twitter.com/HN4lPG96gD— Sancta Familia Media (@sfmmossend) April 30, 2019
Images shared on Twitter of the ransacked church showed damage to the altar area, including broken statues.
The Archdiocese of Glasgow described the incident in a tweet as a "shameful attack on a much loved church".
Anyone that can help with the police's enquiry should call 101 or tweet @CrimestopSCOT.
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