* Negotiators on both sides cite progress on most vexing issues
* Teachers evaluations, test scores, layoffs contentious
* Fourth day of walkout, 350,000 students affected
By James B. Kelleher
CHICAGO, Sept 13 (Reuters) - The head of the Chicago Teachers Union said on Thursday an agreement was near to end a four-day strike in the nation's third-largest school district over education reforms sought by Mayor Rahm Emanuel.
Asked how close the union was to a deal with Chicago school officials, teachers union president Karen Lewis told reporters: "On a scale of 1 to 10, I'm a nine."
But it was unlikely students would be back in school before Monday even if a tentative deal was struck today, she said.
The strike by 29,000 public school teachers and support staff affecting 350,000 elementary and high school students is the biggest strike in the United States this year.
Lewis, who called Emanuel a "bully" and "liar" before leading teachers out on their first strike in 25 years, struck a conciliatory tone for the first time after late-night talks on Wednesday.
Barbara Byrd-Bennett, chief education adviser for Chicago Public Schools, told reporters on Thursday: "Given what happened last night and the progress last night and early this morning with our team ... there's no way - unless we get, like, really nutsy - that we could not have an agreement."
Lewis, speaking separately to reporters before entering talks, said it was unlikely school would resume on Friday because the union's delegates would have to formally approve a deal.
"From the pace at which these things go, I doubt very seriously I can get a House of Delegates meeting together tonight," Lewis said. "Doing something fast is not the way to go ... . I think we still have too much on the table."
As for schools reopening on Monday, Lewis said: "I'm praying, praying, praying - I'm on my knees - for that, please, yes, I'm hoping for Monday. That would be good for us. That would be very good for our kids."
Lewis said there was progress on the two most vexing issues - using student test scores to evaluate teachers and giving more authority to local principals to hire teachers.
The union is concerned that more than a quarter of its membership could be fired because the teachers work in poor neighborhoods where students perform badly on standardized tests, which Emanuel wants to use to evaluate teachers.
"This is really not a 'gotcha' evaluation system," Byrd-Bennett said. "It's to make sure we have a very high standard ... that will keep the very best teachers in front of our students every day."
Lewis said the union fears Emanuel plans to close scores of schools, putting unionized teachers out of work. In recent years about 100 public schools have been closed, with officials usually citing low enrollments. At the same time, a similar number of publicly funded, non-union charter schools have opened.
About 52,000 students enrolled at those schools have not been affected by the strike this week.
The strike in Barack Obama's home city also has put the U.S. president in a tough spot between Emanuel, formerly a top aide to Obama, and the labor unions that Obama is counting on to work for his re-election on Nov. 6.
Obama's own Education Department has championed some of the reforms Emanuel is seeking, and a win for the ambitious Chicago mayor would add momentum to the national school reform movement.
Both sides agree Chicago schools need fixing. Chicago students consistently perform poorly on standardized math and reading tests. About 60 percent of high school students graduate, compared with 75 percent nationwide and more than 90 percent in some affluent Chicago suburban schools.
The fight does not appear to center on wages, with the school district offering an average 16 percent rise over four years and some benefit improvements. Chicago schools already have a projected $665 million budget gap for the year that began in July, a key factor driving Emanuel's reforms.
More than 80 percent of Chicago public school students qualify for free school lunches because they come from low-income households.
"Teachers feel beaten down throughout the country," said Randi Weingarten, national president of the union including the Chicago teachers. "They feel beaten down because of austerity, because of test- rather than teacher-driven policies, because of a spike in poverty, because of the demand on them to do more with less - and then blame them when that doesn't work out."
"That's what's created all the frustration that you hear on the picket line," she said.