Both time and cash are running out, yet plans for solving this city’s financial crisis seem to grow only more tangled by the day. Under a state statute, Gov. Rick Snyder of Michigan must decide by Thursday whether to set in motion a process that would assign an emergency manager to take control of the finances of Detroit, which is mired in $12 billion in long-term liabilities and has said it could run out of money as early as next month. For some here, hopes of avoiding an emergency manager — or, worse yet, plunging into default — are pinned on a consent agreement with the state. This agreement would grant state officials significant oversight of the city’s books and allow Detroit to force cheaper terms in the contracts of unionized city workers by summer as a way to save money. Supporters of the notion, which is comparable to arrangements that have been tried in other states with cities on the edge of bankruptcy, view it as the best choice in an array of miserable options. But others, including those in a crowd that has filled this city’s Council meeting room in recent days, view the agreement as an attack on unions. The city employee unions have met in marathon sessions in the last few months with Detroit officials on contract concessions, only to learn that a new state deal might mean deeper cuts.