RIA Novosti said the center, one of the largest and highest rated in St. Petersburg, will be relocated to create a building for the Supreme Court and Arbitration Court after they move to the city from Moscow.
The decision, publicized late last year, shocked city employees and the public, RIA Novosti said.
Residents said they have rallied against the decision and collected signatures against the move.
Organizers of the protests said they have collected 100,000 signatures in the past few days, and judges and employees of Russia's Arbitration Court have also signed their own petition.
Russian authorities are denying the plans to relocate the center, RIA Novosti said.