Troop levels were on the mind of to Iraqi and U.S. political leaders Tuesday as by Iraqi President Jalal Talabani (search) paid an official visit to the White House. The newly elected leader said, however, that he does not want the U.S. military to set a timetable to withdrawal its troops.

"We will set no timetable for withdrawal. A timetable will help the terrorists, encourage them that they could defeat the superpower of the world and the Iraqi people," Talabani, the first democratically-elected president of Iraq (search) in a half century, said.

Earlier in the day, however, Talibani raised eyebrows when he told news organizations that he could see as many as 50,000 U.S. troops return from Iraq by the end of this year. The remarks were a stark difference from his comments to President Bush (search) in which he said only that he hoped that by the end of 2006, Iraqi forces would be able to take over from their American counterparts.

In a press conference with Bush after a closed-door meeting in the Oval Office, Talibani also thanked the president for U.S. support.

"We will never forget what you have done for our people," he said.

Bush repeated his stance that American troops will stay in Iraq only as long as they are needed.

"As Iraqis stand up, Americans will stand down and when the mission is completed, Americans will come home," Bush said.

During the press conference, Talibani also touched on efforts to confront Al Qaeda and terrorism in Iraq, describing a series of successes against the terror network.

"We have captured many senior elements of Al Qaeda. We have killed many of them. And we had also many of them in our prisons," he said.

Talabani also praised democracy-building efforts in Iraq, which has elections scheduled in October for approval of a draft constitution. Later in a question-and-answer session, the two leaders discussed concern over Syria's lack of cooperation in preventing the movement of foreign fighters across the border between the nations.

The meeting between Bush and Talabani came as the administration's top envoy in Iraq warned that the United States is running out of patience with Syrian interference across the border.

On Monday, U.S. ambassador to Iraq Zalmay Khalilzad (search) refused to rule out either a military strike against Syria or punishment through the United Nations for its lax attention toward the border.

"Our patience is running out, the patience of Iraqis are running out. The time for decision ... has arrived for Damascus," Khalilzad said in a news conference in Washington.

Later Tuesday, Bush left for the United Nations (search) to take part in the annual opening session of the General Assembly. Talabani was also headed to New York for the U.N. meeting.

But Americans seem to have shifted their focus away from Iraq and terrorist threats to issues at home. For the first time since the Sept. 11, 2001, terrorist attacks on the United States four years ago, a majority of Americans responding to a poll by the Pew Research Center last week said it is more important for the president to focus on domestic policy than the War on Terror.

Another poll by Time magazine found six in 10 Americans think the United States should cut back spending on Iraq to help pay for the cost of recovery efforts in the Gulf Coast after Hurricane Katrina. About the same number favor a partial withdrawal of troops from Iraq to help with storm damage.

The president disputed suggestions on Monday that the military is stretched too thin to help both in Iraq and along the Gulf Coast.

"We've got plenty of troops to do both," Bush said after his first on-the-ground tour of cleanup efforts in the streets of New Orleans. "It is preposterous to claim that the engagement in Iraq meant there wasn't enough troops here, just pure and simple."

The president said he will be in "constant touch" with hurricane recovery teams during his two-day trip to the United Nations. Bush's busy schedule there includes one-on-one meetings with the leaders of China, Israel, Britain and U.N. Secretary-General Kofi Annan (search) in addition to General Assembly sessions. Aides arranged to keep him informed of hurricane response efforts at his regular morning briefings.

Russian President Vladimir Putin (search) also was scheduled to visit the White House on Friday after Bush returns from his trip to Louisiana.

"By the time I'm finished [being] president, I hope you'll realize that the government can do more than one thing at one time and individuals in the government can," Bush told reporters Monday as he wrapped up a tour of New Orleans and Gulfport, Miss. "If I'm focusing on the hurricane, I've got the capacity to focus on foreign policy and vice versa."

Fox News' Greg Kelly and The Associated Press contributed to this report.