Al-Umari said Israeli police have evidence linking Olmert to bribery. But Israel has imposed a ban on the release of any information about the charges against the prime minister, and the ongoing investigation.
The police are expected to partially lift this ban by Sunday, he said.
Al-Umari said Tzipi Livni, the Israeli foreign minister, may assume the post of prime minister until the investigation is completed.
"The [foreign] minister will replace him and can keep the job for 100 days. After that the president of Israel nominates people to form a new government by preparing for elections," said al-Umari.
House arrest
Early elections could be held possibly in November to choose a new prime minister.
Police have interrogated Shula Zaken, a director in Olmert's office, for the fourth time. She is currently under house arrest, al-Umari said.
"It seems that Olmert is in a very embarrassing situation. Stepping down is now a matter of time."
The Israeli media have not reported the latest developments in the case, but The New York Post newspaper has carried a story on the alleged bribery scandal, al-Umari said.
Some sources believe that Olmert is being targeted by a right-wing religious group which used to back him when he was with the Likud party as a right-wing activist.
This group, they say, is believed to be pursuing the case in view of possible progress in negotiations with the Palestinians, al-Umari said.