The recruiting by the Bush administration is aimed at relieving the burden of the 140,000 American troops spearheading the occupation force in a country where U.S.-led forces are coming under frequent attack.
Twenty-six other countries already are providing around 20,000 troops. Three of the countries the United States would like to see contribute — Turkey, Pakistan and South Korea — have not yet made a public commitment.
Others have rejected U.S. approaches, such as Brazil. Defense Secretary Donald H. Rumsfeld and Secretary of State Colin Powell have indicated they don't expect a large influx of new international troops, saying contributions may not total more than 10,000.
Washington wants a U.N. resolution giving the world body's authorization to send troops to Iraq — something many countries have said they want before they'll consider joining in. But there are other obstacles.
Some nations are reluctant to send troops because of political differences over Iraq with Washington, internal troubles or a lack of available forces.
India, for example, has armed forces of over 1.3 million, and had been expected to be a major contributor, but Defense Ministry officials say a surge in violence by Islamic militants in Kashmir means Indian troops will be tied down there. Indian officials declined to say how many troops Washington had sought.
Other nations — from supporters of the Iraq war such as Australia and Spain to opponents like Canada and Germany — have ruled out adding to the peacekeeping force.
In Moscow, Defense Minister Sergei Ivanov has said sending Russian troops isn't yet on the agenda. But his failure to rule it out has prompted speculation Russia could participate, partly to ensure a stake in Iraq's oil interests.
A key factor in determining if contributions will come is the U.N. debate on the U.S. draft resolution seeking more money and troops. Critics like France and Germany want assurances that the United Nations would play a stronger role in post-war Iraq and that political power would be turned over quickly from the U.S. occupation administrators to Iraqis.
Turkey is keen to improve relations with Washington and to ensure stability in its southern neighbor. Turkish officials say Ankara is considering a U.S. request for 10,000 to 20,000 troops.
"Lasting peace, calm and stability in Iraq is to Turkey's greatest advantage," Foreign Minister Abdullah Gul said in a television interview last week. "If we send troops, the mission of our soldiers will be to serve this goal."
Another Muslim nation, Pakistan, is also debating a request from the United States and Britain for a division of 10,000 to 12,000 soldiers, officials in Islamabad said.
South Korea is studying a request that would be its biggest overseas deployment since the Vietnam war, during which it sent 320,000 troops to help the United States.
U.S. officials have cited the Polish-led division of 9,500 troops in south-central Iraq as an example for the size of a South Korean contribution, officials in Seoul said.
All three countries, however, face opposition at home to sending the troops.
Any South Korean deployment would likely provoke a fight with anti-war activists, who launched violent protests in April when parliament agreed to send 675 military engineers and medics to Iraq.
In Turkey, a recent poll indicated most Turks oppose sending troops, just as they opposed the U.S.-led war that ousted Iraqi Saddam Hussein. Any deployment must be approved by parliament, which in March blocked plans for Turkey to host American troops intending to open a northern front in the war.
A large contingent of Turkish troops also could provoke unrest from Iraqi Kurds, given decades of fighting between Turkish troops and Kurdish rebels over the border in southeastern Turkey.
New Iraqi Foreign Minister Hoshyar Zebari, a Kurd, has said Iraq's neighbors should not send troops. Other Iraqi officials have said Turkish forces must be deployed well away from the Kurdish territory in the north.
Any Pakistani deployment will face opposition from hard-line Islamic clerics who issued an edict last month saying Pakistani soldiers should not go and any who die in Iraq won't be eligible for martyrdom or an Islamic funeral.
European nations that supported the war say they have few troops to spare.
Britain announced last week it is sending an extra 1,200 troops to reinforce its 7,400-strong contingent. Italy, with almost 3,000 troops in Iraq, and Spain, with 1,300, said they were not considering sending more.
War opponents such as Germany and France rule out participation unless the United Nations and Iraqi authorities get more power. However, they may help indirectly by taking on a bigger share of peacekeeping in Afghanistan, where NATO agreed Thursday to begin planning for a wider role beyond its current mission restricted to the capital, Kabul.
Nicholas Burns, the U.S. ambassador to NATO, urged allies not to turn their backs on Iraq. "I cannot believe that any European who was opposed to the war in Iraq can imagine it's in Europe's interests for Iraq to fail," he said Tuesday.
A bigger NATO role may come later. Diplomats at alliance headquarters say NATO may assume the peacekeeping role now fulfilled by the Polish-led international division of 9,500.
New allies from the old East Bloc are eager to help, but their armed forces are often underpowered and overstretched.
Lithuania has 90 troops in Iraq and may send 50 more. Albania is unlikely to be able to do more than maintain the 70 soldiers it has there already. Romania, which has 600 troops in Iraq, may send 50 more.
Bush telephoned Japanese Prime Minister Junichiro Koizumi on Friday to personally lobby for Tokyo's support in reconstructing Iraq and Afghanistan.
Koizumi won lawmakers' approval in July to send Japan's military to help with reconstruction there, but plans for a dispatch of troops have been on hold over concerns about mounting casualties among peacekeepers.
Citing similar safety concerns, Brazil has turned down a U.S. request for peacekeepers. Officials in Brasilia declined to say how many troops Washington sought from them.