The police and the justice ministry said in a joint statement issued after investigators questioned Olmert for the third time, that "the prime minister was asked to give his account about suspicions of serious fraud and other offences".
"While serving as mayor of Jerusalem and as minister of industry and trade, [Olmert] is suspected of seeking funding for flights abroad in his official capacity from several sources at the same time ... including the state."
The statement also said that each of these sources was asked to pay in full for the same flight.
Police suspect that the "considerable sums" that remained after the flight was paid for "were transferred by Olmert to a special fund [his] travel agency administered for him. These monies were used to finance private trips abroad by Olmert and his family," the statement said
Olmert was mayor of Jerusalem for 10 years until 2003. He later served as a cabinet minister before succeeding Ariel Sharon as prime minister in early 2006.
The case could force Olmert from office and hamper US-backed peace talks between Israel and the Palestinians.
Favours for funds
Authorities are trying to establish whether Olmert dispensed favours in exchange for funds before he became prime minister in 2006.
Al Jazeera's David Chater, reporting from Jerusalem, said: "Police are already talking about a fourth round of interrogations."
"What they're looking at right now [is] an indictment for bribery."
Olmert says he did nothing wrong in his dealings with Morris Talansky, a New York-based Jewish fundraiser, but has promised to step down if indicted.
In testimony on May 27, Talansky told the Jerusalem District Court that he had passed $150,000 to Olmert over a 15-year period, including loans that were never repaid.
Talansky said some of those funds were spent on expensive cigars, hotels and other extravagances.
Olmert says the money was used legally to fund election campaigns.
His lawyers plan to cross-examine Talansky starting on July 17 in court hearings expected to last several days.
The allegations have stirred new calls for Olmert's resignation - and for early elections to be held - from within his Kadima party and its Labour allies.
Olmert averted a coalition crisis over the affair when he agreed to hold a party leadership election.
On Thursday, the Kadima party confirmed that it would hold a leadership election in September.
Tzipi Livni, Olmert's deputy and foreign minister, appears favoured to win, a survey Israel's Yediot Aharonot newspaper showed on Friday.
Shaul Mofaz, Israel's transport minister, and Avi Dichter, the public security minister, are also in the running, although it remains unclear whether Olmert himself will seek to run again.