The man reportedly offered the role of Israeli Defence Minister has called for "disloyal" Arabic citizens of Israel to be beheaded.
Avigdor Lieberman, who has previously served as Foreign Minister between 2009 and 2012 and 2013 to 2015, leads the ultra-nationalist Yisrael Beiteinu party and has been described by some academics as a "far-right extremist".
The controversial cabinet move follows the resignation of his predecessor, Moshe Yaalon, who said he "no longer had faith" in Israel's Prime Minister Benjamin Netanyahu.
Mr Lieberman, who lives in an illegal settlement in the occupied West Bank, said of Arabs in Israel in a 2015 election campaign speech: "Those who are with us deserve everything. Those against us, it cannot be helped, we must lift up an axe and behead them - otherwise we will not survive here."
While Foreign Minister, he called for financial incentives for Israeli Arab citizens to leave the country into a future Palestinian state.
Israeli Prime Minister Benjamin Netanyahu reportedly offered him the choice of becoming the Minister of Defence or Minister of Immigration Absorption, Ynet News said.
Mr Lieberman had initially refused to serve in the current right-wing government in 2015 due to it not being "nationalist" enough.
Professor Juan Cole from the University of Michigan, an expert on Middle Eastern politics, described him as a "far-right extremist", telling the Salon: "There is no European cabinet minister who comes close to Lieberman’s far, far right positions, and if there were he or she would be boycotted by the other Europeans."
He added that in his opinion the current politics of the Israeli government has no real comparison other than in Hungary's neo-fascist party.
The Independent has contacted Mr Lieberman for comment.
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