WASHINGTON — A sharply divided Congress isn’t likely to jump at President Barack Obama’s challenge for quick passage of a mortgage refinancing bill that supporters say could help millions of homeowners save big each year and boost the economy.

Obama praised the legislation in his State of the Union speech last week, saying the proposal would help more homeowners with mortgages backed by Fannie Mae and Freddie Mac take advantage of low interest rates and refinance their loans.

Even with mortgage rates near a 50-year low, Obama said, too many families that have never missed a payment and want to refinance are being turned down.

“That’s holding our entire economy back, and we need to fix it,” the president said. “Right now, there’s a bill in this Congress that would give every responsible homeowner in America the chance to save $3,000 a year by refinancing at today’s rates. Democrats and Republicans have supported it before.”

The proposal is part of a push by Democrats and the White House to help homeowners take advantage of low interest rates as a way to help the housing market recover and to give the economy a shot in the arm.

While the bill could gain traction in the Democratic-controlled Senate, it faces a rough road in the GOP-run House, where many Republicans favor scaling back the government’s role in the housing market as a way of aiding the economy. Similar versions of the measure died in the House and Senate’s lame duck sessions last year.

“At the moment, it’s an uphill battle,” said Rep. Peter Welch, D-Vt., who plans to file the House version of the bill.

Welch said he will reach out to Republicans this year in hopes of building more support