BRASÍLIA—Brazil’s President Michel Temer is under investigation in connection with a sprawling corruption scandal, the country’s Supreme Court said Thursday, as opponents called for his impeachment and financial markets plunged in Latin America’s biggest country.

The high court confirmed the investigation into the president a day after Brazilian newspaper O Globo reported that prosecutors had recordings of Mr. Temer from March purportedly encouraging a corporate executive to keep making payments to silence former Speaker of...