The defection of Peres, 82, would represent a vote of confidence by the Nobel peace laureate in Sharon's oft-repeated pledge to make "painful concessions" for peace with the Palestinians.
Branded a "loser" by political satirists for repeated defeats in national elections, Peres may nonetheless be a vote-getter for Sharon by attracting some of Labour's liberal electorate in Israel's 28 March ballot.
Israeli media reports said Sharon had offered Peres the job of peace envoy if the prime minister's new Kadima party wins the general election, Israeli media reports said.
"They very much want him," Yoram Dori, a spokesman for Peres, said about officials in Kadima.
But Dori said no specific job had been discussed and Peres was "considering what would be the best way to contribute to Israel in the coming years".
Sharon offered Peres a post in a
future government, say reports
"It is a very difficult decision as it is so tied up with historic and other considerations," Peres told reporters on Sunday. "It will take me a day or two to decide."
Peres was surprised at his defeat in a 9 November Labour leadership election to trade union chief Amir Peretz, whose decision to pull the party out of the cabinet reshuffled the political deck in Israel.
"They don't want me in Labour," Peres was quoted as telling confidants.
Sharon and Peres forged an alliance a year ago when Labour joined the Likud-led coalition to support the removal Jewish settlements from the Gaza Strip.
Faced with a far-right revolt in Likud over the Gaza pullout, Sharon this month quit the party he co-founded to capitalise on the broad public popularity of the withdrawal.
Two other Labour loyalists have already defected to Sharon's party, among them a prominent member of Labour's longtime bastion, the kibbutz collective farm movement.
Also on Monday, the brother of Peres caused an uproar when he referred to the new Labour chief's North African origins and said he was a "foreign body" in the party once dominated by Israelis with a European background.
Israelis of Middle Eastern descent have long complained of being marginalised by the Israeli establishment, which is deeply rooted in Europe.
Moroccan-born Peretz defeated
Peres to become the Labour leader
The Moroccan-born Peretz dethroned the Belarus-born Peres in a surprise victory in Labour primaries this month.
Gigi Peres on Monday compared Labour's new chief to the former Spanish dictator, General Francisco Franco, saying Peretz' backers had overrun the party in the manner that Franco's Falange fascists seized control of Spain in 1936.
Peres noted that at one point in his career, Peretz had left Labour to form his own party, One Nation, only to merge with Labour. Referring to One Nation, Peres said that Peretz and his supporters "came over from North Africa, took over and shot them (Labour activists) in the back".
He said Franco drew his strength from the fascists of southern Spain who imposed themselves by force in Madrid and crushed the emerging Spanish republic.
"Peretz and his people are a foreign body in Labour, like general Franco in Spain"
Brother of Shimon Peres
"Peretz and his people are a foreign body in Labour, like general Franco in Spain," Peres told Army Radio.
Israel Radio quoted staff in Shimon Peres's office as saying that the remarks were unseemly and made without his knowledge.
Labour legislator Yuli Tamir, a Peretz ally, demanded that Gigi Peres apologise to his brother, saying he had done Shimon Peres a great disservice, the website of the Haaretz daily reported.