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Vice Premier Shimon Peres is not likely to enter the presidential race, sources close to the minister said following a round of discouraging meetings on Sunday.
Earlier in the day, a Peres adviser said he was leaning toward running.
He said he told Peres his chances of becoming head of state were good, because several MKs had changed their minds and would support him.
Peres has never officially said he would enter the contest, but he told Prime Minister Ehud Olmert last week that he would run. Peres met on Sunday with Acting President and Knesset Speaker Dalia Itzik, also of Kadima, to discuss her taking his place in the race.
Peres aides have been canvassing MKs for weeks, trying to gauge his support among the MKs who will decide the issue. The picture remained murky, however, and the aides told Peres it was not certain that he could win the secret ballot vote.
Itzik, who promised that she was "not interested in filling the position on a permanent basis" in a speech she made after she was sworn in as acting president on January 25, has become increasingly interested in becoming Israel's first female elected head of state.
Meanwhile, MK Colette Avital of Labor vowed to stay in the race despite calls from opponents to withdraw.
Those opponents include Peres adviser Eyal Arad, who called for Avital to drop her candidacy over the weekend.
"I am in the race to win and not in order to pass the baton on to Peres," said Avital. "Everyone has their own considerations. If they say they won't run because of me, it's an excuse."
An official close to Avital said Peres was clearly threatened by Avital's candidacy and the support she had received from Labor lawmakers.
Peres had previously been assured of the support of several Labor MKs who have now announced they will support Avital.
The only declared contenders are Avital and MK Reuven Rivlin (Likud). Tel Aviv Chief Rabbi Yisrael Meir Lau is also considering running, but last week, several female MKs said that if Lau entered the race they would reveal "unpleasant secrets from his past."
MK Shelly Yacimovich (Labor) said she was in touch with women's organization with files on Lau that would embarrass the country "in the vein of the Katsav allegations." President Moshe Katsav is being investigated for alleged sexual and other offenses.
Yacimovich did not give more specific information, but another female MK, who spoke on condition of anonymity, said that they had information relating to inappropriate sexual behavior in Lau's past.
Those MKs, and the women's rights organizations, were criticized over the weekend for using the information at their disposal for partisan political purposes.
Several newspaper editorials asked why it was all right for Lau to serve as chief rabbi of Tel Aviv, but not as president. They called on the MKs to reveal the information if it was of a nature that would prevent Lau from holding "any official post."
Meanwhile, Kadima MKs close to Lau said that he was considering not running, due to the speculation regarding his past. Lau could not be reached for comment.
Gil Hoffman contributed to this report.
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