Bush considers withdrawal timetables to be dangerous war policy that encourages enemies of the United States to wait for troops to leave. The president has also objected to billions in domestic spending that Democrats included in both bills.
The White House has sought to portray Democrats as indecisive, with each passing day coming at expense of the troops.
"One of the things that he's looking forward to hearing is how the Democrats have decided to compromise among themselves first so that he knows what their position is," Bush spokeswoman Dana Perino said just before the president met with lawmakers.
The administration took advantage of a recent congressional break to accuse Democrats of advocating a policy that would signal surrender in the global war on terror, and to demand they send Bush a bill he would sign.
That did little, if anything, to shake Democratic resolve, as demonstrated Wednesday in comments Reid and Pelosi made to reporters after meeting privately with two Iraq war veterans and the relatives of three others.
"We are in ... a war without end," Pelosi said at a news conference. "I hope the president will be receptive to that. In fact, I pray that he will be."
Reid's comments had a sharper edge. He said administration claims that the Pentagon has already begun running low on funds was "all part of the spin coming from George Bush and Dick Cheney, which is baseless."
Also at a news conference, Susan Dinsdale, whose son has served two tours of duty in Iraq, said, "I am begging the president to bring our troops home." Tears welled in her eyes as she added, "This war is senseless. I don't want our soldiers to feel like they have served in vain."
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