President Bush and Russian President Vladimir Putin (search) will sit down and have a few heart-to-hearts this weekend about Iraq's future, Russia's aid to Iran's nuclear program, combating terrorism, the Middle East and weapons of mass destruction.

The two leaders were spending Friday evening and much of Saturday at Camp David (search), the secluded presidential retreat in Maryland's Catoctin Mountains northwest of Washington.

"The president believes this is an opportunity to continue to focus on ways to broaden cooperation between the United States and Russia, as we work in partnership to address shared challenges," White House spokesman Scott McClellan said Friday.

Putin arrived a half-hour late on a sunny fall afternoon. A column of Navy sailors and Marines greeted the two presidents, who walked to face a phalanx of reporters.

"Glad you're here," Bush said as he threw an arm around a smiling Putin.

The two presidents then shook hands with a small group of aides, including White House Chief of Staff Andrew Card, National Security Adviser Condoleezza Rice, Russian Foreign Minister Igor Ivanov and Russian Defense Minister Sergei Ivanov.

Bush drove a golf cart and Putin got in beside him.

Ahead of the meeting, Putin visited the New York Stock Exchange.

He said he hoped for "a breakthrough in our business partnership," even while criticizing U.S. trade restrictions that are vestiges of the Cold War.

Question of Iran

Weapons inspectors from the U.N.'s International Atomic Energy Agency (search) this week found more traces of highly enriched, weapons grade uranium at the Kalay-e Electric Co. This was the second recent finding of the substance.

While Tehran insists its nuclear program is peaceful and used for energy purposes, the United States and some of its allies are highly skeptical.

"You bet I'll talk to President Putin about it this weekend," Bush told reporters Thursday. "It's very important for the world to come together to make it very clear to Iran that there will be universal condemnation if they continue with a nuclear weapons program."

Bush spoke with other world leaders at the United Nations (search) meeting in New York this week about Iran. The IAEA has set an Oct. 31 deadline for Iran to prove that its nuclear program is for energy purposes.

Bush will also address a decade of U.S. complaints about an $800 million deal to build a reactor for a nuclear power plant in Iran. Russia says it still plans to push ahead with the project.

Close Look at Iraq

Bush will also press Putin on the importance of American-led stabilization and reconstruction efforts in Iraq, McClellan said.

Putin opposed the U.S.-led war against Saddam Hussein and joined French and German efforts to prevent passage of a U.N. resolution authorizing the use of force.

During a speech Thursday at Columbia University, Putin indicated that he's leaning toward the French stance on Iraq -- that all services and operations immediately be turned over to the Iraqis and the United Nations -- and said a U.N. resolution calling for more international help won't get much support if it requires continued U.S. control.

Bush administration officials argue it only makes sense for the United States to remain in civilian and military control until all operations can be turned over to local authorities, since it has the most troops in Iraq and led the war in the first place.

But the White House says it envisions a "vital" role for the United Nations going forward.

Secretary of State Colin Powell on Friday said he hopes Iraqis can write up a constitution within six months. Powell is sending a message to the Iraqis, one diplomat told Fox News, "that the Iraqis have to start thinking seriously about a timetable and deadlines."

"We want the Iraqis in charge," said one ally in the sovereignty discussions, "but we also don't want the process to drag on endlessly."

Putin Burying the Hatchet?

Putin has said that Russia is ready to put aside war differences to work with the United States on rebuilding Iraq, even holding out the possibility of eventually sending troops.

But in a speech to the U.N. General Assembly on Thursday, the Russian leader implicitly criticized the United States for launching a war without U.N. approval. "Only direct participation by the United Nations in the rebuilding of Iraq will enable its people themselves to decide on their future," Putin said.

"And only with the active -- and I want to stress this -- practical assistance by the United Nations in its economic and civil transformation, only thus will Iraq assume a new, worthy place in the world community."

Putin has visited Bush at both the White House and his ranch in Crawford, Texas. Washington officials say the Camp David trip reflects the close relationship the two leaders have.

Bush and Putin say they maintain warm personal ties, and there are signs that the U.S.-Russian relationship is bound to grow deeper. For instance, the United States seeks new sources of petroleum, and Russia needs new sources of investment to develop and move its rich reserves.

But points of friction remain.

Deputy Assistant Secretary of State Steven Pifer warned last week that Kremlin policy in Chechnya (search) may be "among the most troubling issues" at the Camp David summit.

Putin has accused the United States of secret talks with rebel representative Zelimkhan Yandarbiyev, who lives in Qatar.

McClellan said that Bush is "not afraid" to raise such touchy issues with Putin.

What Russia Wants

Mark Brzezinski, former director of southeastern European affairs for the Clinton White House, said Putin will likely ask Bush for three things in return for his cooperation on Iraq: a big stake for Russia in the post-Saddam rebuilding of the country, such as more business contracts; less criticism from the west -- the United States in particular -- when it comes to how Russia deals with Chechnya; and normal trade relations with the United States.

"For the first time in U.S.-Russian summetry, a Russian president comes to the United States with more to offer than the U.S.," Brzezinski told Fox News. "The U.S. really politically need Russian involvement ... the Russians know this."

He said the United States has been "very passive" with regards to Russia's activities lately, such as human rights violations reported in Chechnya.

"Because the U.S. is not in a good negotiating position vis-a-vis Russia, the U.S. likely will not seriously raise those as important issues Russia needs to make progress on," Brzezinski continued.

But Bush definitely has some things he can use to up the ante, as well.

"We have plenty of leverage and one is trade and economic development of the country and the other, of course, ie legitimacy," Gen. Alexander Haig, former secretary of state under President Reagan, told Fox News on Friday. "The third is they [Russia] have to now act like a country seeking democratic reform and moving toward a pluralistic society."

The humanitarian aid group Doctors Without Borders made an appeal on behalf of Arjan Erkel, the head of the group's North Caucasus mission, who was kidnapped more than a year ago in the volatile Dagestan region adjacent to Chechnya. The organization, in an ad in The Washington Post, urged Bush to make a personal appeal to Putin on behalf of Erkel, a 33-year-old Dutch citizen.

Fox News' Jim Angle, Wendell Goler, Liza Porteus and The Associated Press contributed to this report.