LOS ANGELES — A newly streamlined government plan to reward homeowners who diligently pay their underwater mortgages is proving a bonanza for banks, which by one estimate may pocket $12 billion in extra revenue by refinancing loans.

The revisions to the Obama administration’s 3-year-old Home Affordable Refinance Program have yielded mixed results for homeowners, analysts and mortgage professionals say.

Some responsible homeowners are indeed getting lower-interest loans despite owing far more than their homes are worth. But others have loans that don’t qualify, or must jump through hoops the plan was supposed to eliminate, such as on-site appraisals and extensive paperwork.

What’s more, critics say, homeowners who get new loans are being stuck with higher rates than necessary, often half a percentage point or more. That’s because banks are refinancing only their own borrowers, instead of competing against one another, which would drive rates down.

“The banks should charge lower than the market interest rate because the new version of the program means less work and less risk for them. Instead, they are charging more,” said Amherst Securities analyst Laurie Goodman, who titled a recent report on the program “And the Winner Is ... the Largest Banks.”

The program is a key part of President Barack Obama’s efforts to bolster the ravaged housing market. Administration officials including Housing and Urban Development Secretary Shaun Donovan are pressuring Congress to pass a law enabling the program to be used to help more homeowners.

“There’s a real urgency here because interest rates today are at the lowest level they have ever been,” Donovan testified Tuesday before the Senate Banking Committee. “But as the economy continues to improve, the expectations are this window of record low interest rates may not last for a long time.”

In response, Sens. Robert Menendez, D-N.J., and Barbara Boxer, D-Calif., said Tuesday that they would introduce legislation this week to extend streamlined refinancing to all underwater Fannie and Freddie borrowers and eliminate appraisal and upfront fees for homeowners using the program to obtain new loans.

The Home Affordable Refinance Program is less controversial than relief plans for delinquent borrowers. Few have objected to its goal of helping homeowners who pay their loans on time but can’t refinance at today’s record low rates because their home values have plummeted.

To qualify, borrowers must owe more than 80 percent of the current home value. They can’t have missed a payment for the past six months and are allowed to have been late by 30 days only once in the last year.

As this year began, nearly 1 million loans had been replaced using the program, but only 1 in 10 had balances higher than 105 percent of the home value. The changes, phased in during the first quarter, aim to encourage refinances no matter how far underwater the loan is.

The program is for loans owned or backed by Fannie Mae and Freddie Mac, the government-supported mortgage buyers that handle 60 percent of U.S. home loans. It works by having mortgage customer-service providers, which are mainly arms of banks, refinance borrowers into new loans that are sold to Fannie or Freddie.

Because Fannie and Freddie already are stuck with the losses if the existing loans go bad, the thinking goes, substituting lower-interest new mortgages actually reduces everyone’s risk. The homeowners have hundreds of dollars more each month, which makes them less likely to default — a boon to their local housing markets and a lift for the economy when they spend their extra cash.

The problem, Goodman said, is that the streamlined program minimizes processing costs for the existing loan servicers but not for competitors, who must collect nearly as much information about borrowers as though they were writing new loans.

The program also exempts existing servicers from having to reimburse Fannie and Freddie for losses on certain flawed mortgages — a multibillion-dollar problem these last few years for the big banks — while requiring competitors to bear that same risk.

Obama envisioned a different scenario when he announced the revised program last fall.

“These changes are going to encourage other lenders to compete for that business by offering better terms and rates,” he said. “And eligible homeowners are going to be able to shop around for the best rates and the best terms.”

That wasn’t the experience of Johnny James, who bought a Gardena, Calif., condominium with a 20 percent down payment during the housing bubble and now owes $414,000 on a home Fannie Mae says is worth $266,000.

James and his wife, Yolanda Hatcher, have full-time jobs with Los Angeles County and excellent credit ratings. Since they hadn’t missed payments on their Fannie Mae loan, they thought they were good candidates for a lower-interest refi.

But their servicer, Seterus Inc., said it was just a bill collector, not a lender. Their original lender, JPMorgan Chase & Co., said it would refinance only loans it is currently servicing. Wells Fargo & Co. said the same, and online mortgage specialist Quicken Loans said the condo was too far underwater to refinance.

“There’s not a lot of help out there for folks like us,” James said.

The couple turned to mortgage broker Jeff Lazerson, who said he submitted applications to eight lenders and found only one that would refinance them. The pending deal, which would cut their rate to 4.63 percent from 6.25 percent, was made after they fully documented their income and assets and paid for an on-site appraisal.

“This program has been billed as a worry-free way for responsible people to get a break on rates even if they’re way underwater,” said Lazerson, president of Mortgage Grader in Laguna Niguel, Calif. “From where I sit, it’s a disaster.”

James Parrott, senior advisor on housing at the White House’s National Economic Council, said that even in its imperfect current version, the program would aid many of the half-million or so borrowers who have applied to refinance since the latest revisions were made.

“Those people get dropped from 6 percent or 7 percent loans to somewhere around 4 percent,” he said. “They will have hundreds of dollars more for themselves every month and thousands of dollars a year.”

While proponents say the program makes winners out of all hands, it is not without detractors.

Alexandria, Va., banking consultant Bert Ely said easy-qualifier loans “are what got us into this mess in the first place” and that waiving legal liabilities for banks could result in another round of mortgage headaches in 2013 and beyond.

“What the government is sanctioning is kicking the can down the road, again,” he said.

Like other administration plans to bolster housing, the voluntary Home Affordable Refinance Program had underperformed until recently. Lenders rarely refinanced loans bigger than 105 percent of the home’s value even though they were permitted to go to 125 percent.

But that changed as the new rules loosened restrictions and did away with the 125 percent cap. Applications for these refinances rocketed from less than 5 percent of the mortgage market in December “to close to 25 percent and rising,” Nomura Securities analyst Brian Foran wrote in a recent report.

The loans are more profitable as well. In the past, Foran said, lenders typically made 2 percent of the loan amount when selling a loan to Fannie or Freddie, so a $350,000 loan might yield $7,000 in revenue.

Because the banks are charging higher than market interest rates for loans made under the program, the mortgages are more valuable to investors and sell for more. The banks are typically making an extra 2 percent of the loan amount, Foran said — another $7,000 on the $350,000 loan, money that drops to the bottom line.

By Foran’s calculations, writing more loans at higher profit could yield $12 billion in additional revenue for lenders.

All the big banks showed unexpected jumps in their first-quarter mortgage profits, in large part because of the revised government program, said Keefe, Bruyette & Woods research director Frederick Cannon.

“Interesting that (the program) would be so good for banks,” he said.