Rumor has it that Chicago officials who have been feverishly negotiating with the Chicago Teachers Union to avert a Monday strike have significantly increased their offer, but union leaders may not bring the offer to members because it doesn’t go as far as they want.

That means that about 29,000 teachers and other school employees could walk out at 12:01 a.m. Monday in the country’s third-largest school district, which has 404,000 students.

The district has set aside about $25 million to keep a number of schools open in the morning so that parents can leave their children there — but teachers won’t be at the head of the classes.

Stay tuned.