The Obama administration would like to keep about 3,000 to 5,000 U.S. troops in Iraq beyond the end of this year but has not begun formal discussions with the Iraqis about the size or makeup of the force, U.S. officials said.

Iraqi Prime Minister Nouri al-Maliki has expressed a desire to keep some U.S. trainers in the country in 2012, past the deadline negotiated by the George W. Bush administration to remove all U.S. troops from the country.

But the Iraqi leader faces staunch opposition from key members of his coalition government who are deeply opposed to any U.S. presence. Some members of the coalition have threatened to boycott the government if it allows any U.S. forces to stay.

Senior U.S. officials have said they are hopeful that they will be able to reach an agreement with the Iraqis on maintaining the small training force. The Iraqis will have to agree to any U.S. presence.

U.S. officials said it is still not clear exactly what kinds of training the Americans would provide and how many trainers would be needed to accomplish the mission. The estimate of 3,000 to 5,000 troops reflects a general consensus on what is politically feasible in Iraq and the United States.

U.S. officials want to keep troops in the country to help shepherd the growth of Iraqi military forces and also to keep an eye on Iran and Syria. Only about 46,000 troops on 47 U.S. bases remain, down from a high of 166,000 on 505 bases during the height of the “troop surge” in 2007.

“You don’t start with troop levels,” said a senior defense official, speaking on the condition of anonymity to discuss sensitive internal matters. “You start with the mission — what they are going to do.”