The officials, speaking on condition of anonymity, said 37 civilians had been killed in the morning clash near Tanhuato in the state of Michoacan, near the border with Jalisco, a region home to Mexico's second-biggest city, Guadalajara.
One official said two federal police had been killed in the shootout, while the other said three had died in Tanhuato, where federal forces replaced local police a week ago following the assassination of a candidate for mayor in a nearby town.
Jalisco, one of the engines of the Mexican economy, is the base of the Jalisco New Generation cartel, which has become a major headache for President Enrique Pena Nieto.
The cartel has killed at least 20 police since March and on May 1 the gang shot down an army helicopter in southwestern Jalisco, killing six military personnel.
In a series of concerted attacks that day, the gang also set vehicles, banks and gas stations ablaze around Guadalajara.
The identity of the gunmen involved in Friday's clashes was not known, but the killings will increase pressure on Pena Nieto to deliver on a pledge he made upon taking office in December 2012 to bring stability to Mexico.
Pena Nieto's Institutional Revolutionary Party hopes defend the slim majority it and its allies have in the lower house of Congress in mid-term elections on June 7.
More than 100,000 people have died in clashes between the drug gangs and security forces since the start of 2007.
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