More than 600 schools could be closed indefinitely, the ASTI warned today as teachers voted on industrial action.
The union was voting on whether to pull out of supervision and substitution work.
They are currently not paid for such work, which includes looking after pupils during break times and filling in for absent staff.
The two-day postal ballot began today and members of the ASTI executive will meet on Saturday to review the result and decide whether to withdraw supervision and substitution.
The Minster for Education Mr Woods is "disappointed" that the ballot does not ask ASTI members about the Government's offer of £27-per-hour for supervision and substitution, a spokeswoman said today.
She said: "He will support any proposals that boards of managers at schools might come up with."
She added the Government would make no further comment until the result of the ballot is known.
Members of the Teachers' Union of Ireland (TUI), have also been balloted over the issue. The results of their postal ballot will not be known for several weeks.
The Irish National Teachers' Organisation (INTO), is continuing negotiations after making some headway on the issue.
PA