Olmert has so far defied a demand by his main coalition partner, Defence Minister Ehud Barak's left-leaning Labour Party, to leave office over a growing corruption scandal.
A poll by Israel's mass circulation Yedioth Ahronoth newspaper found Olmert's deputy, Foreign Minister Tzipi Livni, would win an internal vote to lead their centrist party. Livni, Israel's chief negotiator with the Palestinians, garnered the support of 39 percent of Kadima members, according to the poll. Transportation Minister Shaul Mofaz came in second with 25 percent.
Lawmaker Tzachi Hanegbi, head of Kadima's central committee, told Israel Radio on Friday that Kadima delegates would convene a meeting on a leadership ballot after Olmert returns from a visit to the United States at the end of next week.
Kadima sources said Olmert wants his centrist party to put off any such vote for months, hoping to ride out the police investigation into allegations he accepted envelopes filled with cash from a Jewish-American businessman.
Olmert has denied wrongdoing but has said he would resign if indicted. The turmoil threatens to derail U.S.-backed peace talks between Israel and the Palestinians.
Sixty percent of Kadima's members believe Olmert does not have to resign at this stage in the investigation, the Yedioth Ahronoth poll found.
Olmert has responded to the crisis with a business-as-usual approach. He is expected to meet Palestinian President Mahmoud Abbas as early as Sunday,
Israeli officials said. Palestinian negotiator Saeb Erekat said a date has yet to be set.
Israel's attorney-general, Menachem Mazuz, said on Thursday the investigation would be accelerated "in order to complete it as soon as possible". He gave no precise timeframe for a decision on whether to indict the prime minister.
Olmert has survived similar scandals in the past.