WASHINGTON — America and Iraq have agreed to seek “a general time horizon” for deeper reductions in American combat troops in Iraq despite President Bush’s once-inflexible opposition to talking about deadlines and timetables.

Iraqi officials, in a sign of growing confidence as violence decreases, have been pressuring America to agree to a specific timeline to withdraw American forces. The White House said today that the timeframe being discussed would not be “an arbitrary date for withdrawal.”

Mr. Bush and Prime Minister Maliki talked about the timing issue as part of discussions over a broader security agreement to keep American troops in Iraq after a U.N. mandate expires on Dec. 31.

The White House says the two leaders, in a conversation yesterday, agreed that the accord should include “a general time horizon for meeting aspirational goals, such as the resumption of Iraqi security control in their cities and provinces and the further reduction of U.S. combat forces from Iraq.”

Iraq has proposed requiring American forces to fully withdraw five years after the Iraqis take the lead on security nationwide — though that precondition could take years to meet.

Earlier this month, Iraq’s national security adviser, Mouwaffak al-Rubaie, said Baghdad would not accept any security deal unless it contains specific dates for the withdrawal of American-led forces. Mr. Maliki also has said he expects the pending troop deal with America to have some type of timetable for withdrawal.

Mr. Bush has vetoed legislation approved by the Democratic Congress setting deadlines for American troop cutbacks.

The White House statement said the timing of further reductions would be linked to improved security conditions. In recently weeks, Iraq’s government has expressed increasing confidence about its authority and the country’s improved stability.

America has handed control of 10 of 18 provinces to Iraqi officials.

The White House tried to make a distinction between talking with Iraqis about withdrawals and attempts by Congress to force cutbacks.

“I think it’s important to remember that the discussions about timeline issues previously were from Democrats in Congress who wanted to arbitrarily retreat from Iraq — without consideration for conditions on the ground,” the White House deputy press secretary, Scott Stanzel, said.

“All of the discussions that we have always had have been based on conditions on the ground and making progress in the country and we are doing just that. We are making progress on the security situation,” he said. “The number of attacks has dropped dramatically in recent months.”

The American military buildup in Iraq that began more than 18 months ago has ended. In recent days, the 3rd Infantry Division’s 2nd Brigade, the last of the five additional combat brigades sent in by Mr. Bush last year, left the country. There are still 150,000 U.S. troops in Iraq — as many as 15,000 more than before the buildup began.

The chairman of the Joint Chiefs of Staff, Admiral Mike Mullen, said on Wednesday that he is likely to recommend further troops reductions this fall because security has improved.

“I won’t go so far as to say that progress in Iraq from a military perspective has reached a tipping point or is irreversible — it has not, and it is not,” Admiral Mullen said. “But security is unquestionably and remarkably better. Indeed, if these trends continue I expect to be able early this fall to recommend to the secretary and the president further troop reductions.”

Defense Secretary Gates said officials are looking for ways to send additional American troops to Afghanistan this year. A cutback in Iraq could clear the way for more troops to go to Afghanistan.

The White House statement said the reduction in American forces in Iraq was “a testament to the improving capacity of Iraq’s Security Forces and the success of joint operations that were initiated under the new strategy put in place by the president and the prime minister in January 2007.”

___

Reporter Deb Riechmann contributed to this report from Tucson, Ariz.