Netanyahu will have up to six weeks, beginning Thursday, to form a majority coalition, Xinhua news agency reported.
He is facing struggles as his right-wing Likud faction won 32 seats in last Tuesday's voting, far cry from the needed 61 seats to form a majority coalition in the 120-seat parliament, or Knesset.
President Reuven Rivlin on Wednesday said in broadcast remarks that he tapped Netanyahu with the task of forming the country's 36th government although Benny Gantz and his centrist Blue and White party won a narrow 33-32-seat victory over Netanyahu's Likud.
Rivlin said his decision followed a two-day official consultations process, during which 55 lawmakers said they supported Netanyahu as the next prime minister and agreed to join a governing coalition under his leadership.
Gantz had 54 lawmakers recommending him but 10 of them, lawmakers with the Arab-Jewish Joint List, said they will not join a possible governing coalition under Gantz.
Netanyahu said he accepted the appointment and called for a "wide unity government" under his leadership. Gantz rejected the call.
Gantz, a former military chief, said he is "committed" to the idea of unity but will not join forces with a leader who might face serious corruption charges, referring to Netanyahu's involvement in several corruption cases. The attorney-general has said he intends to indict Netanyahu, pending a hearing.
Gantz said that during two days of talks on a possible unity government, brokered by President Reuven Rivlin, the Likud representatives said that any future government should include the pro-settler party of Yemina and two ultra-Orthodox parties.
"This composition would not allow a wide government under my leadership that will work for the benefit of the entire citizens of Israel," Gantz said.
Rivlin implored all of the heads of the parties to show "responsibility" and strive for a unity government, saying it was the only solution to break the political deadlock. "The people do not want a third election," he said.
The elections on September 17 were the second time Israelis cast their ballots in five months.
After Netanyahu had won a narrow victory over Gantz's Blue and White in the April's election, he failed to form a governing coalition. Instead of risking Rivlin handing Gantz the chance to form a government, Netanyahu called for a snap election.
Coalition-building in Israel usually drags on as smaller parties demand cabinet seats and financial and legislative changes to fulfill the promises they gave to their voters during the election campaign.
( With inputs from IANS )