If the bill eventually becomes law, Arabic would be defined as having special status, while "its speakers have the right to language-accessible state services," though it would not be an official language, Haaretz newspaper reported.
Hebrew is defined as the "national language" in the bill, which would become part of the country's so-called basic law, which is similar to a constitution, it said.
Ministers confirmed the proposed legislation had been approved by a cabinet committee, allowing it to move on to parliament.
Some 17.5 percent of Israel's population are Arab. Public signs and government services are typically in Hebrew as well as Arabic and it was unclear whether the new bill would change that.
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Prime Minister Benjamin Netanyahu's government is seen as the most right-wing in the country's history, and Arab Israelis allege widespread discrimination.
Defining Israel as the "national home of the Jewish people" has also raised concerns among rights activists and others worried over discrimination and attempts to further mix religion and state.
However, Haaretz said the most recent version of the bill would not subordinate democracy to Israel's Jewish character, unlike previous versions.
Parliament member Ayman Odeh, who heads the mainly Arab Joint List alliance, said approving the bill would mean trampling on minority rights, adding it would "legally transform us into second-class citizens".
The bill was sponsored by Avi Dichter, a member of Netanyahu's Likud party who said it was necessary to "set in law our national identity while remaining a democratic state".