Tel Aviv — Prime Minister Olmert of Israel resigned after 33 months in office and six corruption probes, setting off a process in which he will remain caretaker while Foreign Minister Livni tries to establish a government.

Mr. Olmert presented his resignation letter late yesterday to President Peres, who holds the official task of determining which party leader is most capable of forming a stable coalition government. Ms. Livni, Olmert’s successor as chief of the ruling Kadima Party, will likely get the nod.

“This evening, Prime Minister Ehud Olmert submitted a letter of resignation from his post,” Mr. Peres said in a short statement broadcast from his residence in Jerusalem. “I appreciate the respectful way in which he is handing over his power.”

Mr. Olmert, 62, succumbed to pressure to give up the premiership that mounted as police conducted one criminal investigation after another against him and ultimately recommended September 7 that he be indicted for fraud, bribery, and money laundering. He has denied any wrongdoing.

Mr. Olmert announced during the weekly Cabinet meeting earlier in the day that he would resign, saying, “This is not an easy decision.” He did not speak publicly after presenting the resignation letter to Mr. Peres, who said he will probably make his decision tomorrow on whether Ms. Livni should be given the job of forming the new government. Mr. Peres leaves late today to attend the annual meeting of the United Nations General Assembly in New York.

With his resignation, Mr. Olmert formally becomes a caretaker prime minister who will run the government until Ms. Livni can prove she can take over and maintain a stable majority in the 120-seat Knesset. Kadima, which has 29 seats, controls a total of 64 in parliament through its alliance with the Labor, Shas, and Pensioners parties.

The decision to withdraw from political activities by the transportation minister, Shaul Mofaz, who was close runner-up to Ms. Livni in the September 17 Kadima primary, may hurt Ms. Livni in her coalition-building attempts as it makes her look weaker in her own party, a professor of political science at Hebrew University, Avraham Diskin, said.

“It complicates matters for her,” Mr. Diskin said in a phone interview. “It is more dangerous in both the short and long run because people may leave Kadima when the time comes and join Likud.”

Ms. Livni, seeking to form a broad government, has offered Likud leader Benjamin Netanyahu the post of foreign or finance minister, the daily Ma’ariv said yesterday, without citing anyone.

If Ms. Livni doesn’t retain the support of the existing Cabinet or assemble a new coalition within 42 days of being nominated by Mr. Peres, then national elections could be held within months.