The vote came amid increasing tensions between Israel and the Palestinians, with a wave of deadly attacks on Israelis and clashes at protests in the occupied territories.
Israel's parliament, the Knesset, voted unanimously in favour of the bill in first reading on Wednesday, with about half of all deputies in attendance.
It will now be studied in committee before returning to parliament for a second vote.
Both Arabic and Hebrew are official languages of Israel, but while the vast majority of Israeli Arabs speak Hebrew, Arabic is not widely spoken among the Jewish population.
The bill was introduced by lawmaker Oren Hazan, of Prime Minister Benjamin Netanyahu's rightwing Likud Party, who said it was meant to reach out to Arab Israelis.
"Language is a door to culture," he told AFP. "I am looking reality in the eyes and I understand there is no possibility to walk to peace without understanding each other."
Arab Israelis, the descendants of Palestinians who remained on their land after the creation of Israel in 1948 and who are citizens of the Jewish state, make up about 18 percent of its population.
However they often complain of discrimination.
Hand in Hand, a centre for Jewish-Arab Education in Israel, welcomed the vote, with its chief executive Shuli Dichter calling it "a good and important decision".
'BASIS FOR LIVING TOGETHER'
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"Daily communication is the basis for living together in a shared society. However, it is only the first step. Knowing the other side's culture and its core narratives are no less important for achieving that end."
The surge in violence in recent weeks has seen nine Israelis killed in attacks.
Many of the attackers have been shot dead, including one in Hebron on Wednesday. Along with those killed in anti-Israeli protests, 60 Palestinians have died in the unrest, as well as one Arab Israeli.
Some have compared the wave of violence to the two previous Palestinian intifadas, or uprisings, the last of which took place in the early 2000s.
Members of both the Jewish and Arab communities say they are fearful of the other.
Hazan said the classes could help ease such concerns.
"In the last few months the intifada is rising in Israel," he said.
"When people stand in the bus station and they hear (Arabic) next to them, automatically... they maybe think that somebody is going to do something bad to them -– just because they don't understand."
He pointed out that the ancestors of many Jews in Israel, himself included, came from the Arab world.
One potential sticking point may be around ultra-Orthodox Jewish schools which have a different curriculum to mainstream Israeli schools.
However Hazan said every official school in Israel would be bound by the law.