Chicago Mayor Rahm Emanuel is preparing to press for a property-tax increase of about $500 million to shore up police and firefighter pensions that threaten the city’s solvency, Bloomberg News reported on Friday. The proposal will be part of Emanuel’s Sept. 22 spending plan for the budget year beginning Jan. 1. The increase, expected for months, would be the centerpiece of a budget that is $426 million out of balance. The city faces a financial reckoning after years of failing to save enough to pay the benefits it promised employees. Over the past decade, Chicago has put $7.3 billion less into the pension funds than actuaries recommended. Its next annual pension payment is projected to jump 10 percent, to $976 million. Chicago’s effort to reduce its liabilities hit an obstacle in July, when a judge ruled the benefits cuts it sought were illegal. The city will appeal to the Illinois Supreme Court, which in May threw out a pension overhaul adopted by the state, saying workers’ pensions are protected.