FLINT, Mich. — Michigan officials announced charges Tuesday against two former emergency managers who were appointed by the state to oversee this financially troubled city, the latest criminal charges to grow out of a monthslong investigation into Flint’s tainted water.

The emergency managers — Darnell Earley and Gerald Ambrose — were charged with false pretenses, conspiracy to commit false pretenses, misconduct in office, and willful neglect of duty in office. Also charged Tuesday were two former Flint workers, Howard Croft and Daugherty Johnson, who are accused of false pretenses and conspiracy to commit false pretenses.

Officials in Flint showed “a fixation on finances and balance sheets,’’ said Bill Schuette, the state’s attorney general.

“This fixation has cost lives,’’ he said. “This is all about numbers over people, money over health.’’

The charges against the emergency managers are likely to renew debate over Michigan’s practice of appointing such managers as a way to shore up cities that are teetering on the edge of fiscal disaster. There are systems of state-over-locality oversight in about 20 states, but Michigan’s system has drawn sharp criticism and was even the subject of a referendum calling for its repeal by voters in 2012.

Before Tuesday, nine people — including eight low- and mid-level state workers and an employee of Flint’s water plant — had been charged in the water contamination, which has been linked to the lead poisoning of Flint children and the deaths of 12 people from Legionnaires’ disease.

The contamination here began after the city switched its water supply to the Flint River from Lake Huron in 2014 to save money. Flint has struggled financially, and it was under the oversight of a state-appointed emergency manager when it made the decision to change water sources.

Officials failed to properly treat the new water with chemicals that would prevent materials from corroding and leaching metals like lead, and as months passed, officials discounted or overlooked a growing body of evidence that something was wrong with the water, e-mails and testimony have suggested. For months, officials played down the reports of Flint residents, who said that their tap water had puzzling, murky colors and vile smells, and that they were feeling ill or suddenly suffering from rashes.

E-mails show that some officials were aware of a possible connection between Legionnaires’ cases and the new water supply at least 10 months before Governor Rick Snyder issued a warning to residents. State officials have defended their response, and a spokeswoman for Snyder said the governor told the public as soon as he learned of the possible link.