"I am convinced that the right man to be the next president of the country is our friend Shimon Peres," Barak said, despite expectations in some political circles the two could clash over peace moves.
"I am convinced he will be an excellent president. I intend to work for his election," Barak told their Labour Party group in parliament.
Army Radio said that Peres had accepted the candidacy for presidency from the Labour Party.
"Mainly, my priority will be national unity. Just as I devoted myself to other things, this time I will devote myself to mediation between the different sectors," Peres said in remarks on Israel Radio.
"I believe that what happened in the past year, since Ehud was elected prime minister, is that internal differences narrowed, really narrowed, and there is a real chance to unify the nation ... and that will be my main occupation."
Weizman, threatened with impeachment over a gift-taking scandal, will resign from the largely ceremonial job on July 10, three years before the end of his term. Officials said the 120-member parliament could elect a replacement on July 31.
Peres's likely opponent is former right-wing Likud party cabinet minister Moshe Katzav.
Israeli media have long alluded to a personality conflict between the 76-year-old Peres, a former Labour leader who lost five elections, and Barak, his 58-year-old successor and Israel's most decorated soldier.
Now head of a Middle East peace centre that bears his name, Peres was due to meet Barak and declare his candidacy later on Monday.
Israel's state prosecutor last week decided not to charge Weizman with a crime for accepting more than $300,000 in gifts from a French millionaire from 1987 to 1993. But public outrage forced the hand of the 75-year-old former air force chief.
Weizman, who as a politician helped forge Israel's landmark 1979 peace treaty with Egypt, was elected president in 1993 and won a second term in 1998.
Weizman personified the sharp-tongued native Israeli "sabra," who was tough on the outside but soft on the inside. But many felt the latest scandal, one of several to dog Israeli politicians of late, was too much.
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