Olmert's visit to Washington gave Bush a chance to redirect his attention to foreign policy after his Republican party's losses in congressional elections.
There has been speculation in the US that Bush, weakened by the election results and setbacks in Iraq, might try to complete his term in office with progress on the Israeli-Palestinian issue.
Olmert has appeared confident that there will be no change in the relationship between the US and Israel.
Olmert told reporters on the flight to Washington: "Support for Israel traditionally has been bipartisan. I don't see anything changing in the next two years that can alter the balance of feeling toward us."
The US and Israel accuse Iran of developing its uraniam enrichment programme in order to acquire nuclear weapons, although Iran says it intends to use the programme purely for electricity generation.
Before his return to Tel Aviv, Olmert had previously told reporters: "If someone wants to reach a compromise with Iran, he must understand that Iran won't be ready to do so unless it is afraid.
"Israel has various options which I am not prepared to discuss."
Iran warned on Sunday that its Revolutionary Guards would respond swiftly if Israel attacked it.
Mohammad Ali Hosseini, an Iranian foreign ministry spokesman, said: "If Israel takes such a stupid step and attacks, the answer of Iran and its Revolutionary Guard will be rapid, firm and destructive and it will be given in a few seconds."
Olmert said in an interview broadcast on Israeli television on Monday: "I am not looking for wars. I am not looking for confrontations. I'm looking for the outcome.
"This campaign will be tested in only one way - whether it will succeed to stop Iran from possessing nuclear weapons."
Speaking at the Simon Wiesenthal Tolerance museum in Los Angeles shortly before his departure on Wednesday, Olmert said: "We will do what needs to be done [to prevent Iran from acquiring nuclear weapons].
"I am happy to say after meeting president Bush that he shows the same attitude, the same conviction and the same commitment."