A Bavarian cabinet minister said the aim was to protect school pupils against fundamentalist influences.
The bill is expected to be passed next year in the regional parliament, which is dominated by the Christian Social Union (CSU) party.
Bavaria, after Baden-Wuerttemberg, is the second state to propose a ban.
Bavarian Education Minister Monika Hohlmeier said the headscarf was increasingly used as a political symbol.
“With this law, we are defending pupils against a potential fundamentalist influence and are respecting the wishes of the majority of parents,” she said.
Christian and Jewish symbols are not included in the ban.
Ms Hohlmeier said these symbols reflected cultural values.
The plans are similar to a draft law unveiled in neighbouring Baden-Wuerttemberg last month.
They follow a controversial ruling in September in Germany’s highest court.
Thirty-one-year-old Fereshta Ludin, who was denied a job in Baden-Wuerttemberg in 1998 because she wore a headscarf in school, went to courts arguing that the German constitution guaranteed her religious freedom.
The federal Constitutional Court ruled by five votes to three that, under current laws, she could wear the scarf. But it also said new laws could be passed by individual states.
Civil rights organisations and groups representing the 3.2 million Muslims living in Germany have strongly criticised the proposed ban.
They argue that the right to wear a headscarf is a question of religious freedom.
A group of prominent German women published an appeal last week not to ban Muslim headscarves.
The women, including the government’s integration advisor Marieluise Beck, argued that the ban might in fact set back efforts to emancipate Muslim women, by effectively keeping them from participating in public life.