"I can say... that, to my knowledge, and on the basis of what I know and read, I believe the efforts of the international community will succeed, and that Iran will not become a nuclear power," he told the Maariv daily.
"There is an enormous effort on the part of the international community to prevent Iran from becoming a nuclear country. Israel plays an important part in those efforts, without leading them."
Building on that, and in an allusion to recent threats made against Iran by an Israel minister, Olmert added: "That is why Israel should not resort to threats as made recently."
Last week, Israeli National Infrastructure Minister Benjamin Ben-Eliezer warned that any Iranian attack against Israel "would lead to the destruction of the Iranian nation."
That prompted a response from the deputy commander of Iran's army, Mohammad Reza Ashtiani, that his country would "eliminate Israel from the global arena" if it were attacked by the Jewish state.
Olmert's remarks were published as President Mahmoud Ahmadinejad said at a military parade marking Army Day that "Iran is the most powerful and independent nation in the world."
Ahmadinejad said all the branches of the armed forces would react forcefully in response to any attack against Iran, and boasted that no one would dare to launch a strike on the country.
The United States and Israel, the Middle East's sole if undeclared nuclear power, accused Iran of using its civilian nuclear power programme as a cover for attempting to develop an atomic bomb.
Tehran vehemently denies that, but has had three sets of United Nations Security Council sanctions slapped on it over its refusal to stop enriching uranium.