Some 30,000 extra US troops have been deployed to boost security
Senate Majority leader Harry Reid, a Democrat, said the president's troop "surge" strategy was not working.
He urged those high-profile Republicans who have broken with President George W Bush in recent weeks to vote with Democrats to end the war.
The administration is to report to Congress on Iraq progress by 15 July.
The interim report on 18 measures of progress is required by law under a previous funding bill. Bush administration officials say the picture will be mixed.
Reports quoting unnamed officials have said it will conclude that the Iraqi government has failed to meet any of the political and economic benchmarks it has been set.
Meanwhile, Iraqi Foreign Minister Hoshiyar Zebari has warned that a premature withdrawal of US troops from Iraq could trigger a civil war and wider Middle East conflict.
The US military commander in Iraq, General David Petraeus, says fighting the insurgency there is a long-term endeavour which could take many years.
Gen Petraeus and US ambassador Ryan Crocker will give a progress report in September on the strategy, under which some 30,000 extra US troops have been deployed in Iraq.
Support for Mr Bush is wavering just as the Senate discusses the military spending bill.
Among those to defect are high-profile Republican Senators Pete Domenici and Richard Lugar, a senior Republican on the Foreign Relations Committee.
According to the New York Times newspaper, White House officials who previously thought they could wait until September to begin discussions on withdrawal from Iraq now think Mr Bush must act quickly to forestall more unrest in his party.
Democratic Senator, Jim Webb, spoke of his frustration.
"We're now in the fifth year of ground combat operations in Iraq, and this deck of cards is coming crashing down.
"And it's landing heavily on the heads of the soldiers and the marines who have been deployed again and again, while the rest of the country sits back and debates Iraq as an intellectual or emotional exercise," he said
The White House has denied claims it is contemplating pulling US troops out imminently.
"But the idea of trying to make a political judgement rather than a military judgement about how to have forces in the field is simply not true," Mr Snow said.
He also sought to lower expectations ahead of the interim report on progress in Iraq, saying it was unrealistic to expect all the benchmarks to have been met already.
Democratic leaders may hope to capitalise on the growing divisions in the Republican ranks to make progress on their demands for a timetable for withdrawal from Iraq.
One amendment, sponsored by Democrat Jim Webb and Republican Chuck Hagel, would require troops to spend the same amount of time at home as in combat, making extended deployments more difficult.
Another proposed amendment, by Democratic Senator Carl Levin, would require that troop withdrawals begin within 120 days of the bill being passed.
Gen Petraeus told the BBC's John Simpson that more time was needed to see the results of the surge because all the troops had only been in place since mid-June.But with US casualties mounting and questions over how much progress is being made, many US lawmakers are under pressure from constituents to pursue a change of strategy.