The Consulate on its website cited "the ongoing demonstrations in the city" for its decision. It said the facility will also remain closed Wednesday because of a local holiday.
The announcement said other U.S. consulates in India and the U.S. Embassy in New Delhi would remain open for consular services.
The U.S. Consulate in Chennai asked U.S. citizens traveling in and around the city "to exercise caution and monitor local media for security updates."
Anti-U.S. protests in front of the consulate continued for the third straight day in Chennai, the Press Trust of India news agency reported, adding another group planned to hold a demonstration Tuesday.