A leadership vote would give Netanyahu an opportunity to win a new mandate from his party after his failure to secure a parliamentary majority in national elections in April and last month left him politically weakened.
In a statement, the party spokesman said Netanyahu was weighing the idea "to shatter the illusion of a Likud rebellion" against him and persuade any political rivals who might be anticipating one that they should join a unity government with him now.
Political commentators said victory in a Likud leadership election could also pre-empt a scenario in which Israel's president, who tasked Netanyahu last week with forming a government, might choose another member of the party to do so if the prime minister does not succeed.
Maariv Online news website said Likud's Central Committee would convene next Thursday to set a date for a party election.
"A snap primary ... will unite (the party) around my leadership," the website quoted Netanyahu as saying at a meeting of Likud legislators.
One potential Netanyahu challenger in Likud, former cabinet minister Gideon Sa'ar, tweeted "I am ready", after the party vote was mooted. But other prominent Likud members swiftly voiced their loyalty to Israel's longest-serving leader.
Netanyahu has called for a power-sharing deal with his strongest political rival, centrist Blue and White party leader Benny Gantz. Gantz also emerged from the Sept. 17 election with no clear path to a governing majority in parliament.
Gantz has said his party would not forge an alliance with Likud with Netanyahu at its helm, citing looming corruption indictments against the prime minister, who has denied any wrongdoing.
At a swearing-in ceremony in parliament on Thursday for legislators elected last month, Netanyahu said Israel's security challenges made a national unity government imperative and a new general election would be damaging to the country.
"Let's form a government quickly that will act for the good of all Israeli citizens," Netanyahu said in the speech, making no mention of a Likud ballot.