The House measure is aimed at addressing the intense anger lawmakers and millions of Americans have with how funds from the $700 billion Troubled Asset Relief Program, or TARP, approved late last year have been spent.
Critics in Congress and elsewhere, including many leading economists, say the first half of the program had little affect on underlying problems in the economy.
A key criticism is that banks failed to use funds to expand lending for consumers and businesses, while billions of dollars went to help ailing U.S. automakers and failing companies.
The legislation is designed to strengthen accountability and transparency in the original legislation and close loopholes.
In addition to requiring that $100 billion of the second half of the program address rising home foreclosures, the measure aims to help expand consumer credit.
It contains a reporting requirement for new lending by the program under President Barack Obama. One amendment requires a report to Congress on how money has been spent so far.
While Republicans acknowledge the measure contains some good new provisions, they question whether it will be effective.
Republican Jeb Hensarling asserted Wednesday that the House risked sending what he called "the wrong signal" to the new Obama administration. "We are essentially saying to the incoming administration, 'Here is a 350-billion dollar bank account.' Well I say, 'Where is the plan?'," he said.
House financial services chairman Democrat Barney Frank noted that a Senate vote last week approving the release of the second half of the funds means that the Obama administration already has authority to spend it. "They know they have the $350 billion. This is an effort to strengthen our hand when we impose some constraints on them," he said.
The House vote sends the legislation to the Senate for possible consideration there.
Senate Majority Leader Harry Reid commented Wednesday on efforts to make the TARP program more accountable: "We have a letter from the President of the United States who [stated] every decision he will make as to how this money is going to be spent. It will be transparent. There will be oversight," he said.
At his Senate confirmation hearing on Wednesday, President Obama's choice for Treasury Secretary, Timothy Geithner, said the TARP program must be adjusted. "We have to reshape and redesign this program to ensure there is credit available to support recovery. We will do this with conditions to protect the taxpayer and the necessary transparency to allow the American people to see how and where their money is being spent," he said.
At the same time, Geithner called the financial institution rescue program essential to U.S. economic recovery.