Senate Republican leader Mitch McConnell meanwhile said the chamber appeared ready to pass a bill "without a surrender date."
But there was immediate anger from the anti-war wing of the Democratic Party.
"I cannot support a bill that contains nothing more than toothless benchmarks and that allows the president to continue what may be the greatest foreign policy blunder in our nation's history," Democratic Senator Russ Feingold said.
Democratic leaders upped the ante in talks with the White House last Friday, again insisting on the need for a date for troop withdrawals to begin — though offering Bush the power to waive the requirement.
But all along, they said they would get a bill funding the troops until the end of September to the president's desk before leaving on a week-long recess tomorrow.
There was no immediate reaction from the White House, which earlier cited the need to keep ongoing negotiations under wraps.
At least 15 US servicemen were reported killed this weekend in Iraq, and 76 so far this month, bringing total US losses since the invasion to 3,419, according to an AFP count based on Pentagon figures.
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