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“This bill doesn’t go far enough in my opinion, so I had to hold my nose,” Mr. Griffith said.

He had voted against the first House bill last week, arguing it granted too big a debt increase, and said he actually liked the two-step process in the new House bill because it would force the debate on spending to continue.

Mr. Griffith said the bill is now a message to the Senate, where attention shifted late Friday.

In the vote, six Republicans joined all 53 members of the Senate Democratic Caucus in voting to table the House plan. Those Republicans argued it didn’t cut deeply enough, while Democrats said it didn’t last long enough.

They are looking for a debt increase that will go through at least the 2012 election.

President Obama on Friday made his own appeal for a new bipartisan deal to emerge from the Senate.

“There are plenty of modifications we can make to either of these plans in order to get them passed through both the House and the Senate and would allow me to sign them into law. And today I urge Democrats and Republicans in the Senate to find common ground,” the president said.

Mr. Obama has been on the sidelines for most of this week after Mr. Boehner cut him out of talks.

Still, the president said the sides are closer than it would seem. All parties agree that the deal will not include tax increases, and that any debt increase must be matched dollar-for-dollar by new reductions in future spending.

Mr. Obama said he would even welcome a trigger to kick in a future commission to propose more deficit reduction.

But he insisted any debt increase be all at one time, and be big enough to last through the 2012 election, so that Congress doesn’t go through this debate again. Mr. Obama also has opposed requiring a balanced budget amendment, as House Republicans are insisting on.

And Democrats mocked the new balanced budget requirement in the bill, with one of them saying the language was drafted so poorly that it would allow any amendment with the title that includes the words “balanced budget” would apply.

Rep. Jared Polis, Colorado Democrat, said that meant he could take an Equal Rights Amendment and rename it the Balanced Budget Amendment and it would satisfy the requirement.

“I think we will seriously look at re-titling that amendment,” Mr. Polis said.