There appears to be no easy agreement in sight in a battle between the city teachers union and the Department of Education over what to do with teachers who are on the city payroll but not in full-time teaching spots.

The debate kicked off this spring when a nonprofit group reported that the teachers cost the city $81 million between 2006 and 2007. A second round launched this week when the same group, The New Teacher Project, projected that the teachers will cost the city $74 million this year alone.

The report led the union and the department to relaunch private discussions about the issue — and led each to issue public letters with their most detailed proposals yet on how to resolve it.

Both sides say the pool is a waste of taxpayer dollars, but they differ sharply on how to drain it.

The pool of teachers, known as the Absent Teacher Reserve, was created as a result of the 2005 teacher contract, which required that both teachers and principals consent to every placement of a teacher in a school.

The result is that teachers whose positions are eliminated have no guaranteed spot at a school, and they remain on the city payroll without a full-time position unless a principal agrees to hire them.

To deal with the glut — which grew this year by more than 700 teachers, to a total of more than 1,000 — each side is proposing a different solution.

The president of the city teachers union, Randi Weingarten, is proposing that the city halt all hires of new teachers until the teachers in the reserve pool are placed.

Mr. Klein is instead proposing that the city remove teachers who are in the pool if they do not find a placement after a certain period of time.

Neither side is prepared to budge. Ms. Weingarten is calling Mr. Klein’s proposal an abandonment of the job security she negotiated for as a condition of allowing the 2005 contract’s open-market principles.

She is also saying that Mr. Klein’s idea would hurt the city’s efforts to attract quality teachers to difficult schools.

Many of the reserve teachers lost their positions when their schools were shut down by the city, and Ms. Weingarten said that allowing their jobs to be cut could send a bad message to possible teachers.

Mr. Klein is fighting back, pointing out that Ms. Weingarten’s recommendation to freeze new hires would create a situation in which a principal could only choose his new hires from the reserve pool.

Mr. Klein is saying that tying principals’ hands in this way would be a return to the pre-2005 contract, in which teachers were forcibly placed in schools according to seniority, whether prinicpals wanted them or not.