The City of Detroit, straining to manage its debts and meet its payroll, may be nearing circumstances dire enough to require a state-appointed emergency financial manager, Michigan officials cautioned city leaders this week.

Detroit, once the nation’s fourth-largest city but one where the population has slipped to 713,000, has long grappled with a shrunken tax base and other financial woes. A year ago, city officials found themselves in similarly dismal circumstances — at risk of running out of cash — and in April agreed to a consent deal that allowed the city to operate with help from an oversight board that includes officials chosen by the state.

But critics say the oversight board has proved unable to help overhaul Detroit’s finances swiftly or drastically enough. Adding to the troubles, Mayor Dave Bing and members of the Detroit City Council have clashed over how best to turn the city around, even as the threat of having to furlough city employees looms. And complicating matters, in November Michigan residents voted to overturn the state’s emergency manager law, which granted sweeping power to state appointees sent to struggling cities and which was a legal underpinning of the agreement that brought the oversight board to Detroit.