Some 33,000 children, mostly Muslim Arabs, attend the schools, which have a history predating Israel's foundation in 1948 and are run primarily by the Roman Catholic church.
Traditionally, the schools received 65 percent of their budgets from the state, with parents paying the balance. But that figure was cut to 34 percent two years ago, doubling the amount parents had to come up with.
Father Abdelmassih Fahim, director of schools for the Catholic church's Custody of the Holy Land, said the situation had become untenable and he is pressing for a return to the 65 percent funding figure.
"It is a matter of equality," he told AFP.
"A Jewish Israeli child had the right to 100 percent (of school costs financed by the state) while our schools don't, while our teaching is among the best in Israel."
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Fahim said the authorities are proposing that the Christian schools move from being "unofficially recognised" to "officially recognised," effectively meaning their buildings and staff would come under state control.
"We categorically reject that," he said.
The office of President Reuven Rivlin has said he and Education Minister Naftali Bennett met with church officials this week, and that the "president welcomed the important work of these schools".
Fahim said "these declarations are positive, but we are now awaiting the minister's proposal."
"If we do not get satisfaction, we will be obliged to announce that the school year will not begin on Tuesday."
The talks are taking place at a time when Christians in Israel are under growing strains, with leaders of their communities saying they are afraid in the wake of attacks by Jewish extremists on churches and other properties.