WASHINGTON — President Bush yesterday tied his fresh push for Middle East peace to the fight against terrorism and American efforts to counter Iran’s quest for greater influence in the region.

“As we saw on September the 11th, 2001, dangers that arise on the other side of the world can bring death and destruction to our own streets,” Mr. Bush said in his weekly radio address. “Since then, extremists have assassinated democratic leaders from Afghanistan to Lebanon to Pakistan. They have murdered innocent people from Saudi Arabia to Jordan and Iraq.

“They are seeking new weapons and new operatives so they can attack America again, overthrow governments in the Middle East and impose their hateful vision on millions.”

In his radio broadcast, Mr. Bush briefly sketched the agenda for his eight-day trip to the Middle East, which begins Tuesday, the same day as the New Hampshire presidential primary.

Mr. Bush is visiting Israel and the Palestinian Arab territories, plus Arab allies Kuwait, Bahrain, United Arab Emirates, Saudi Arabia, and Egypt. He said he will encourage Israelis and Palestinian Arabs to make “tough decisions on complex questions” so an elusive peace deal could be reached.

“I am optimistic about the prospects,” Mr. Bush said.

His advisers, however, have all but ruled out a three-way meeting with Israeli and Palestinian Arab leaders during the trip, dampening any thoughts that the president’s personal diplomacy would yield a concrete peace accord at this time.

Mr. Bush said he will urge Arab leaders to support negotiations between the Israelis and Palestinian Arabs and stress the “importance of countering the aggressive ambitions of Iran.”

The president argued that success in Middle East peace is crucial to success in the battle against extremists, to whom the violent, intractable Israeli-Palestinian Arab dispute is a potent recruiting tool.

“I know it is not always obvious why events in the nations of the Middle East should matter to the American people,” Mr. Bush said. “But in the 21st century, developments there have a direct impact on our lives here.”

Mr. Bush’s series of bilateral meetings begin Wednesday in Jerusalem with Prime Minister Olmert and President Shimon Peres. On Thursday, Mr. Bush travels to the West Bank, an Israeli-occupied Palestinian Arab territory, to see President of the Palestinian Authority, Mahmoud Abbas, and Prime Minister Salam Fayyad at their headquarters in Ramallah.

Before leaving Israel on Friday for Kuwait, Mr. Bush will also meet with a former British prime minister, Tony Blair, now the Middle East representative for the so-called Quartet — the United Nations, the European Union, Russia, and America.