In the one-page message, distributed by e-mail and in some cities door by door, the gang claims it will halt all crime activity during January to demonstrate that the cartel "is not responsible for the criminal acts federal authorities are reporting to the media."
Prosecutors have not verified the letter's authenticity, according to an employee of the Michoacan bureau of the federal Attorney General's Office who spoke on condition of anonymity because he is not authorized to discuss the matter publicly.
The government says La Familia has been weakened by a recent string of arrests and deaths of top leaders.
In another letter that circulated in November, La Familia purportedly offered to disband.
Last month, gunmen torched vehicles across Michoacan and used them as barricades to block all entrances into the state capital of Morelia after federal police killed alleged La Familia leader Nazario Moreno Gonzalez.
La Familia has occasionally made public pronouncements seeking to convince the public that it is defending Michoacan against other drug gangs.
Federal officials, however, say the cartel has terrorized the state with kidnappings, extortion, hundreds of murders, decapitations and drug trafficking.