Some analysts say that Benjamin Netanyahu, Likud's leader, has a better chance of building a majority in the 120-seat parliament, with the third-placed party Yisrael Beitenu likely to lend him support.
"We have this strange phenomenon in which an election took place and there is no real winner but, having said this, I think that finally with [Yisrael Beitenu leader] Avigdor Lieberman or without his support, Netanyahu will be the next prime minister of Israel," Gideon Levy from Israel's Haaretz newspaper told Al Jazeera.
"Until the very last minute it will not be clear what he [Lieberman] decides ... but I think that finally, after all the games, it will be Netanyahu."
Rightist parties made sweeping gains in the election, which was held just over three weeks after Israel led a 22-day war on the Gaza Strip.
Peres has said he will only make a decision on who he will appoint prime minister after holding talks at his residence with members of all the parliamentary parties.
"I intend to consider public sentiment and the outcome of the elections, and to decide only after consulting with all of the Knesset [parliament] factions," he said on Monday.
Kadima on Sunday suggested a power-sharing deal with Likud but Netanyahu has dismissed the idea.
However, he has said that he favours having Kadima in a broad coalition of parties rather than just organisations that are likely to be supportive of Likud's stance on the Palestinians.
Livni has said that she wishes to continue holding US-backed peace talks with the Palestinian Authority (PA) and that Israel must be prepared to give up the occupied West Bank as part of a final status deal.
However, Yisrael Beitenu and other rightist parties in the Knesset remain opposed to pulling out of the territory and are supportive of settlement expansion there, despite it being contrary to international law.