Peres is under pressure from political allies to seek the premiership, according to officials in his office. For now, they say, he has no plans of stepping down from his largely low-key, ceremonial post.
The overtures to Peres reflect the Nobel laureate's late-career star power and the dearth of viable challengers in Israel's fragmented opposition to Prime Minister Benjamin Netanyahu.
Hopes for a Peres candidacy are "an act of desperation," said political scientist Avraham Diskin. "It shows that there is a large vacuum in the center. They're unable to join ranks, so they're looking for miracles."
Netanyahu, already riding high in the polls, recently tried to solidify his bid for re-election by joining forces with his foreign minister, Avigdor Lieberman.
Running on a joint list, Netanyahu's Likud Party and Lieberman's Yisrael Beitenu are expected to win more than 40 parliamentary seats. That reflects their separate strength in the polls, not more, but together they would form by far the largest faction in the 120-seat parliament and put Netanyahu in a strong position for another term as prime minister.
The partnership has fueled calls for centrist parties to band together.
Polls show a unified bloc of the Labor, Kadima, Yesh Atid and Independence parties could rival Netanyahu's hard-line bloc. A poll in the Yediot Ahronot daily Friday found that a center-left bloc plus parties representing Israel's Arabs could win 59 seats, almost enough to block Netanyahu. The poll by the Dahaf agency questioned 500 people and had a margin of error of 3.8 percentage points.
While the centrist parties share similar ideologies, particularly a softer line toward peacemaking with the Palestinians, their leaders have shown little interest in unifying. Peres could be one of the few figures capable of rallying these parties behind him. He also may be best positioned to do so.
Ex-Prime Minister Ehud Olmert, another potential heavyweight candidate, is in the midst of a bribery trial and has not decided whether to seek office again.
Peres brings to the table a resume that is unmatched.
After more than six decades in politics, he is the ultimate political survivor with the gravitas to take on Netanyahu. Winner of the 1994 Nobel Peace Prize, he mingles among the world's rich and famous, receiving rock star welcomes at conferences and ceremonies around the globe. He has a good relationship with both the Palestinians and the White House.
As president, he has cultivated an image as a sage, grandfatherly figure, calming the nation in his deep, monotone Polish-accented voice during times of trouble. He has also courted younger Israelis, embracing social media and backing high-tech industries.
An official in Peres' office said that over the few weeks, a number of operatives with ties to centrist parties, as well as retired military officials, approached him about leading a unified bloc. The official declined to identify the people who had spoken to Peres and spoke on condition of anonymity because the discussions were private.
While describing the pressure as "heavy," the official said Peres had no intention of leaving the presidency, saying he was committed to finishing the remaining two years of his term.
Messages to Labor and Kadima were not returned, while officials with Yesh Atid and Independence said they were not trying to enlist Peres.
Eyal Megged, a columnist at the liberal Haaretz daily, implored the country's opposition politicians to line up behind Peres.
"Peres is the only man capable of slightly deflating the pumped-up egos leading the political camps to the left of the religious-right bloc. He is the only one whom none of those egos would presumably feel insulted to stand behind," he wrote.
If Peres returns to party politics, that image he has built as president could quickly crumble.
For years, Peres had a reputation as a nasty but losing political operator, due to his inability to win an election. Of his three previous stints as prime minister, two were in a caretaker capacity and a third was in a power-sharing arrangement after elections ended in stalemate.
Only late in life, at 83, was he elected president in a parliamentary vote. He even lost an earlier run for the ceremonial post in a stunning upset.
While Peres is said to be in excellent health, his age would certainly become a factor.
The apparent yearning for Peres stems partly from the dire state of peace efforts with the Palestinians, and a sense that time is running out to reach an agreement.
The Palestinians seek an independent state in the West Bank, east Jerusalem and Gaza Strip, areas captured by Israel in the 1967 Mideast war.
Israel's centrist and dovish parties support broad withdrawal from the West Bank, and perhaps part of east Jerusalem, to make way for a Palestinian state. Israel withdrew from Gaza in 2005.
Netanyahu, while endorsing the concept of a Palestinian state, opposes the broad territorial concessions that most believe would be required for peace.
The opposition parties have largely avoided the Palestinian issue in the campaign so far. Instead, they have focused on domestic economic issues, while paying lip service to reviving the deadlocked peace talks.
Peres, who has a good relationship with Palestinian President Mahmoud Abbas, would likely move quickly to get peace efforts on track. He has been one of the few voices during Netanyahu's tenure to speak of the importance of pursuing peace with the Palestinians.
"One thing is for sure: If Peres returns to political activity, no one is going to be able to ignore the most important issue to the future of Israeli society anymore," columnist Emmanuel Rosen wrote in Yediot Ahronot.