BAGHDAD, Iraq – Iraq’s prime minister said yesterday he wants American troops “on their way out” as soon as his government can protect its new democracy. The top American general in the country said he hopes to begin significant withdrawal by next spring.

At the same time, in an unannounced visit, Defense Secretary Rumsfeld said Iraqi security forces should take on more tasks now performed by American troops.

American military commanders have repeatedly expressed hopes in recent months that they could begin major troop reductions next year, depending on the intensity of the insurgency. Even so, yesterday’s remarks seemed to signal a new willingness to discuss specific ways American troops might exit a war in which nearly 2,000 have died.

There was a subdued reaction in Congress. The chairman of the Senate Armed Services Committee, John Warner, a Republican of Virginia, said: “I remain concerned about establishing timetables and raising expectations. However, I have not seen the data that the general had before him.”

Senator McCain, of Arizona, the second-ranking Republican on the committee, said he agreed with General George Casey and Prime Minister al-Jaafari that withdrawal will be possible only when conditions permit.

“I’m sure that the security situation at the time will dictate what they need to do,” Mr. McCain said.

Mr. al-Jaafari, speaking at a joint news conference with Mr. Rumsfeld, said, “The great desire of the Iraqi people is to see the coalition forces on their way out.”

However, comments by the most senior commander of coalition forces in Iraq, General Casey, drew the most notice. He told reporters that a “fairly substantial” withdrawal of American troops could go ahead in the spring and summer of 2006 if the Iraqi political process is not derailed and the insurgency does not grow.

Pentagon officials have provided little detail in discussing the possible withdrawal of forces from Iraq. The most specific estimate has come from Lieutenant General John Vines, who runs day-to-day military operations in Iraq. He said in June that a reduction of “four or five brigades” – perhaps 20,000 troops out of the current 135,000 – was possible sometime next year.

The Bush administration has refused to set deadlines for withdrawal, saying American soldiers will remain as long as needed until Iraqi troops and police can defend the country on their own.

“The president wants to see our troops come home, but we’ve got an important mission that we need to complete,” a White House spokesman, Scott McClellan, said yesterday.

Mr. McClellan said President Bush depends on his commanders to say how many troops they need, “And they make decisions based on the conditions and the progress that’s being made on the ground.”

Car bombings, roadside bomb attacks, and other violence against American forces and Iraqi civilians have increased in recent months, and Mr. al-Jaafari and his cabinet face constant threats. Mr. Rumsfeld’s visit to Iraq was not announced ahead of time for fear of attack.

Still, America points to successful elections in Iraq in January and formation of a democratic government as evidence both that the Iraqis are on the path to independence and that the American-led toppling of Saddam Hussein was worthwhile.

Mr. Rumsfeld did not comment on Mr. al-Jaafari’s remarks, but later told reporters that the country’s political leaders need to do more to relieve the burden on American forces. He said Iraqis need to start taking responsibility for guarding the estimated 15,000 American-held prisoners, and should meet the August 15 deadline for completing a draft constitution.

“It would be very harmful to the momentum that’s necessary” if the constitution is not finished on time, he said. “We have troops on the ground there. People get killed.”

“We desire speed in that regard,” Mr. al-Jaafari said of an American pullout, but added that no specific timetable exists for a handover.

“We do not want to be surprised by a withdrawal that is not in connection with our Iraqi timing,” he said, speaking through a translator.

At the Pentagon, officials say they have prepared a range of scenarios, from increasing the size of the force to reducing it substantially. Drawdowns could be accomplished by bringing home troops ahead of schedule, or rotating in fewer replacements when a unit returns to its home base.

General Casey and other officials mention spring or summer as possibilities because the American military will need time to assess the effects of the December elections on the insurgency, defense officials said. Iraq’s security forces are also expected to be in better shape by then.