As state lawmakers try to zero in on a budget deal, some are predicting a big hike in aid to local schools.

The Republican-led Senate has proposed a $1.9 billion increase (about 9 percent); the Democrat-led Assembly $1.8 billion. Gov. Andrew M. Cuomo has proposed $1.1 billion if lawmakers make changes to teacher evaluations; $377 million if they don’t. Some legislators predict the final number will be north of $1 billion.

But another education trade off might be struggling to gain traction. A Cuomo proposal to tie an education tax credit favored by private schools to college aid for children whose parents entered the country illegally faces opposition in both houses.

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