Under a proposal laid out by Mayor Bloomberg today that took education insiders by surprise, the city would retain access to threatened federal dollars for struggling schools by riffing on a familiar strategy: school closure.

The announcement in today’s State of the City address sets the stage for a showdown with the United Federation of Teachers — and maybe also with the State Education Department.

UFT President Michael Mulgrew had already dismissed the idea that schools could receive the funds without union support by this afternoon. But State Education Commissioner John King has yet to weigh in on the strategy.

Under Bloomberg’s plan, the city would swap dozens of schools from one federally mandated overhaul strategy to another in a bid to escape a requirement that the city and union come to terms on a new teacher evaluation system. An impasse over negotiations caused King last week to cut off federal funds to 33 city schools that were undergoing the “transformation” and “restart” strategies, which require new evaluations.

Under the mayor’s plan, the schools would undergo “turnaround” instead. Turnaround is more aggressive than the other strategies, requiring at least half of a school’s teachers to be replaced. But it also does not require that new teacher evaluations be in place, according to the Obama administration’s guidelines for the funds, known as School Improvement Grants.

Mulgrew immediately dismissed the plan, arguing that the union would have to sign off on turnaround. That would be true — but only if Bloomberg had been talking about the type of turnaround that the Obama administration envisioned.

What the city is actually proposing is using a second, lesser-known turnaround that state regulations allow. Essentially, the city would close 33 schools and reopen them immediately, with new names and identification numbers. Then a team of educators selected for the “new” school would hire a new staff with the union’s input, pulling half of the new teachers from the original school’s roster.

The process is based on a piece of the current union contract known as Article 18-D, which outlines a hiring process used whenever the city closes a school and replaces it with others. Article 18-D requires that at least half of the original school’s staff stays on.

The only difference between the city’s longstanding school closure policy and today’s turnaround proposal is that the students at the turnaround schools would not be displaced. This distinction is significant, given criticism from state officials and others that the city’s closure policy has turned some schools into dumping grounds for high-needs students.

It’s not clear whether the state’s close-and-reopen approach will pass muster with the Obama administration, which has a stricter standard for turnarounds and has urged states to collaborate with their teachers unions.

But it has gotten the state’s endorsement in the past. In September, King signed off on the city’s application for SIG funds to support 11 long-planned school closures, earmarking federal money 16 replacement schools.

Those schools kept receiving SIG money even after King cut off the federal funds to transformation and restart schools, a promising sign for the strategy the city announced today.

King did not respond publicly to Bloomberg’s speech today. But he has spent the last month talking tough on evaluations and is unlikely to be enthused that the city seems to have discovered a way to avoid negotiating them, at least temporarily.

The city’s solution to the evaluations impasse is stopgap at best. Settling on new evaluations is required for the city to receive other pots of federal funds or to negotiate a new contract with the United Federation of Teachers.

King could have some recourse against the city’s proposal. When he froze the city’s SIG funds, he told school officials that they had to notify the state if they wanted to seek approval to change the overhaul model they were using. It’s conceivable that he could deny the request, leaving the city stuck using strategies that require an evaluation deal to fund.

Also, because turnaround wouldn’t start until September in the schools, it’s unclear whether the city could expect this year’s federal funds to be restored, even if King does give his okay.

The city will have little wiggle room on execution if it gets a green light to move forward with the turnarounds. State education law requires the city to post “Educational Impact Statements” for proposed school closures at least six months before the first day of the school year when they would begin. That means the city has just weeks to craft the statements for 33 schools — more than it has ever proposed to close in a single year before. Any missteps would provide material for a legal challenge from the union.

Plus, the combination of the Obama administration’s requirement that no less than half of teachers be replaced and the union’s requirement that no more than half of teachers be replaced means that the reconstituted schools would have to achieve a perfect 50 percent balance between existing and new teachers.

Teachers who are not selected for the new schools would enter the Absent Teacher Reserve, the city’s pool for teachers without permanent positions. Bloomberg has blamed the reserve for costing the city millions of dollars a year.

Several principals could also be out of their jobs. Even under the second type of turnaround that Bloomberg is pursuing, the model requires that principals who have been at their schools for more than two years be replaced.

The news arrived to principals at the 33 schools today amid a tumultuous year. In recent weeks, they were left waiting to see what would come of federal grants they had been promised as teacher evaluation negotiations disintegrated.

One principal, who asked not to be identified, said she was told today that she would be able to keep her job. But, she said, “If I have to do this, I would leave on my own. I would never fire 50 percent of my staff.”

Under the city’s proposal, all of the 14 schools currently undergoing restart, which hands control to nonprofit managers, would become turnaround schools and keep their nonprofit partners. So would most of the transformation schools, 13 in all. And the city would add six schools, including several it had decided against closing this year, to the roster of those receiving SIG funds.

Six schools currently undergoing transformation would stop receiving SIG funds. The city has already announced plans to close two of the schools, and it said today that it would find new funds to finish out the year at two other schools that have not responded sufficiently to improvement efforts.

Two other schools would continue to overhaul their programs but would not receive federal funds to do so, city officials said. The two schools, Chelsea Career and Technical Education High School and Boys and Girls High School, have longtime principals who would likely have to be removed under turnaround.