Governors of more than 30 other states, according to reports, have received similar letters demanding that state chief executives leave office within three days.
Federal investigators do not see threats of violence in the letters but are concerned they could prompt others to respond violently.
Nixon spokesman Scott Holste said Friday a letter sent by the "Guardians of the Free Republics" was received earlier this week.
Holste referred further questions to the Missouri State Highway Patrol. A spokesman there says the agency is investigating with federal authorities and declined to comment about the letter's contents.
The "Guardians of the Free Republics" says on its Web site that it wants to "restore America" by peacefully dismantling parts of the government.
The letters come less than a week after a Christian militia group,, was raided by the FBI. Hutaree planned an elaborate attack on local and state police, according to U.S. Attorney Barbara McQuade.