The board voted 8-1 Tuesday night not to open the schools, scheduled for Aug. 8, The (Memphis) Commercial Appeal reported.
The city has not approved the district's budget, as required by Tennessee law, and the district must submit it to the state by Aug. 1.
"We've been patient. We've cut 1,500 jobs," said board member Tomeka Hart. "… We're not saying give us everything you owe. We are just saying we have to have the money in the bank from our city so we can pay our bills."
City Council President Myron Lowery said the funding issues are currently tied up in court and therefore not negotiable.
"The council supplies less than 10 percent of almost a billion-dollar school budget," he said. "They have voted to delay for having less than 10 percent in hand. That is ridiculous."
Earlier, Mayor A.C. Wharton told the council, "The money is in the budget -- no ifs, ands or buts about it. ... We have fully funded Memphis city schools for the fiscal year 2012."