Mr Peres is expected announce his decision within the next few days
Israeli President Shimon Peres is meeting the heads of Israel's two main political groups to assess which should be asked to form the next government.
Centrist candidate Tzipi Livni's Kadima party won most of the votes by a slim margin in an election on 10 February.
But Benjamin Netanyahu, leader of the right-wing Likud party, has the support of more parliamentary factions.
Mr Netanyahu's bid for PM was boosted on Thursday when he won the backing of the far-right Yisrael Beiteinu party.
Yisrael Beiteinu leader Avigdor Lieberman said he wanted to see a "wide" coalition government taking in the three largest parties.
Ms Livni has indicated she would prefer opposition to a unity government.
Mr Peres is expected to announce his decision on who should form a government in the next few days.
Both Mr Netanyahu and Ms Livni failed to make impressive gains in Israel's election and, says the BBC's Katya Adler in Jerusalem, Mr Peres is thought to want them to work together.
Poll-weary Israelis fear whatever the make-up of their future government, it risks being a weak, unstable one, adds our correspondent.
Since the polls, Mr Peres has been holding consultations to determine who to task with forming a government.
This is typically the party with the most seats, which would be Kadima, which beat Likud by 28 seats to 27.
But nationalist and religious parties, natural allies of Likud, also did well in the election.
With the support of Mr Lieberman, Likud can in theory now count on the support of 65 of the Israeli parliament's 120 members, putting it in a stronger position to form a government.
If Mr Netanyahu is invited to form a government he will have six weeks to sew up a coalition.
Mr Lieberman told Mr Peres on Thursday that he supported the Likud leader as prime minister but "only as part of a wider government" including Kadima and Yisrael Beiteinu, Israel Army radio reported.
Mr Netanyahu won the support of a leading far-right party on Thursday
"Netanyahu will be prime minister, but it will be a Bibi-Livni government," said Mr Lieberman, using Mr Netanyahu's widely-used nickname.
Kadima campaigned on election pledges to continue peace talks with the Palestinians, while Mr Netanyahu stands further to the right and has opposed the negotiations.
Ms Livni's party had proposed a power-sharing deal with Likud similar to the one in 1984 which saw the leaders of Likud and Labour each hold the post of prime minister for two years.
The 10 February election was called after Prime Minister Ehud Olmert handed in his resignation in September after being questioned by police over a series of corruption scandals. He has denied any wrongdoing.
Mr Olmert has stayed on as acting premier.