The ban, the first in Scandinavia, targets the niqab as well as burqas, balaclavas and masks, and would apply in nurseries, schools and universities.
However headscarves, hats and caps could continue to be worn.
Most parties support the bill, which is expected to pass next year.
"These clothes prevent good communication, which is important for students to receive a good education," Minister of Education and Research Torbjorn Roe Isaksen said in a statement.
