President Barack Obama made his pitch today for a broad refinancing plan that he said could save more than 10 million homeowners $3,000 a year.

As he outlined in his State of the Union address last week, Obama’s plan would aid homeowners who have stayed current on their mortgages but cannot refinance because their homes have dropped too much in value.

Obama acknowledged in his speech in Falls Church, Va., that his earlier efforts “didn’t work at the scale we’d hoped.”

In what appeared to be a veiled reference to Republican presidential frontrunner Mitt Romney, Obama said, “it is wrong for anyone to suggest that the only option for struggling, responsible homeowners is to sit and wait for the housing market to hit bottom. I refuse to accept that, and so do the American people.”

The new plan also has the political advantage of pitting responsible homeowners against the biggest Wall Street firms, which would have to foot the bill.

Congress has repeatedly blocked Obama’s efforts to charge a fee on big Wall Street banks to pay for the financial firms’ bailouts, so now he’s proposing to use a portion of the fee to pay for something far more popular.

The White House said the cost of the refinancings would be $5 billion to $10 billion, even though they would have the potential to refinance trillions of dollars of mortgages.

Obama cautioned that his plan would only aid “responsible” homeowners.

“This plan, like the other actions we’ve taken, will not help the neighbors down the street who bought a house they couldn’t afford, then walked away and left a foreclosed home behind. It will not help those who bought multiple homes just to speculate and make a quick buck.”

Obama also proposed a “homeowners’ bill of rights” aimed at cleaning up abuses in the mortgage market and a new program to convert foreclosed properties into rental housing.

Like much of the rest of the president’s agenda, Obama’s housing agenda faces a difficult road getting through Congress. Republicans quickly sent out releases pointing to a series of underperforming Obama housing programs over the past three years.