BRASÍLIA—Brazil’s Supreme Court voted Thursday to put Senate President Renan Calheiros on trial for alleged embezzlement, adding to political turmoil that threatens to engulf the fledgling administration of President Michel Temer.

In an 8-3 vote, the high court chose to indict Mr. Calheiros, a fellow member of Mr. Temer’s Brazilian Democratic Movement Party, or PMDB. Prosecutors say Mr. Calheiros falsified his Senate expense accounts, using fake receipts to obtain reimbursement for 6.400 Brazilian reais ($1,843) a month for...