"A probe is currently underway in Dresden, headed by the Saxony Criminal Police Department and with the participation of special forces. Searches are underway in other cities too", Saxony police tweeted on Wednesday morning.
Five properties in Dresden and one in the nearby town of Heidenau were raided in a probe of suspected preparations for a serious act of violence.
According to the police, the investigation was opened after plans to assassinate Kretschmer were discussed in a group chat on the popular instant messaging app Telegram.
The chat, dated 7 December, was initially uncovered by journalists working for the broadcaster ZDF.
The prosecutor said ZDF had revealed the contents of messages allegedly involving a hundred members of the chat group "linked by their opposition to vaccines, to the state, and the current health policies".
The chat featured audio messages in which people urged to oppose COVID-19 measures "with armed force if necessary".
Germany has faced large protests against restrictions imposed during the COVID-19 pandemic, including in Saxony, which is one of the regions worst hit by the virus, and where the vaccination rate is lower than the national average.