Homeowners who have resisted the urge to refinance their mortgages  until now could be rewarded for their willpower. Mortgage rates have  fallen to new lows—and banks are rolling out incentives to win business.

Economic uncertainty in Europe and slow growth in the U.S. are  prompting investors to pile into ultrasafe U.S. Treasurys. That, in  turn, is pushing down mortgage rates, which are tied to Treasurys.

The average interest rate on a 30-year mortgage fell to 4.05% for the week ended Dec. 23, the lowest in 60 years, according to HSH  Associates, a mortgage-data firm. And rates on jumbo mortgages—private  loans that in most parts of the country are larger than $417,000—also  have hit new lows, averaging 4.61%.

“It’s hard to argue rates will get much lower than they are today,”  says Stuart Gabriel, director of the Ziman Center for Real Estate at the University of California, Los Angeles.

That’s good news for homeowners. A person who refinanced a $400,000  30-year mortgage in February would pay an interest rate of 5.04% on  average, according to HSH Associates, and fork over $2,157 a month; at  the current rate of 4.05%, he’d save $236 per month, or $2,830 per year.

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What’s more, demand for refinancing is declining, since many  homeowners already took advantage of lower mortgage rates. Applications  for refinancing are 17% below this year’s peak in September, according  to the latest data from the Mortgage Bankers Association.

That and other factors have prompted some lenders to offer incentives to win new business—particularly regional and community banks, which  are focusing more on jumbo mortgages, says Stu Feldstein, president at  SMR Research, which tracks the mortgage market.

The discounts can be sizable. Regional bank Valley National Bank  charges homeowners in New Jersey and eastern Pennsylvania a flat fee of  $499 for closing costs on mortgages as large as $1 million. Since  average closing costs on a refinance run about 2% of the total loan  amount, a person with an $800,000 mortgage could save about $15,500.

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A spokesman for the bank says it is aggressively marketing the discount in part to bring in more customers.

While many lenders don’t refinance mortgages that are larger than  about $2 million, Union Bank—which has branches in California, Oregon  and Washington—refinances up to $4 million at no extra cost. (Many banks that refinance multimillion-dollar mortgages tack up to an extra  quarter of a percentage point on the interest rate.)

Since November, Union Bank has also allowed borrowers to roll the  costs of a refinance, like the appraisal fee and loan processing fee,  into the mortgage. And borrowers whose original mortgage is from Union  Bank don’t have to provide all of the income documentation that other  customers do in order to refinance.

In part, the bank’s goal is to develop relationships with  high-net-worth clients, says Stuart Bernstein, national production  manager of residential lending at Union Bank.

Despite the incentives, many would-be applicants remain sidelined because they can’t meet the long list of qualifications.

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The home-equity requirement is one of the toughest hurdles, says Mr.  Feldstein. Homeowners with at least 10% home equity make the cut, but  people with less have a tougher time.

Borrowers with 10% to 19% equity in their home usually have to buy  private mortgage insurance, whose cost varies based on many factors,  including their credit score. A borrower with 15% equity and a FICO  credit score above 720 could pay 0.44% of the total loan amount, says  Keith Gumbinger, vice president at HSH Associates. On an $800,000 loan  that would be $3,520 a year—eating into the potential savings of a  refinance.

In December, the federal government rolled out a revamped version of  the Home Affordable Refinance Program with relaxed home-equity  requirements, to allow more borrowers to refinance. To qualify, the  current mortgage must be owned or guaranteed by Freddie Mac or Fannie  Mae, and borrowers need to be mostly current on payments.

For regular refinancing, applicants need a FICO credit score of at  least 740 to get the best rates, says Mr. Gumbinger. And they must  provide copious documentation, including at least two years’ worth of  tax returns and proof of income as well as recent statements for assets  such as retirement and brokerage accounts.

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After clearing those hurdles, you might wait about 60 days for  refinancing to be completed, says Mr. Gumbinger—longer than the typical  45 days. While some lenders are offering 60-day rate locks for free,  others charge a quarter of a percentage point of the total loan amount  for the service. On an $800,000 mortgage, that’s $2,000.

Or you could opt to take your chances with a free 45-day lock and  hope rates don’t spike between day 46 and the date your loan closes.  With the euro zone still in economic crisis and global investors rushing to the safety of U.S. Treasurys, housing-market analysts say it could  be at least six months before rates rise significantly.

Write to   AnnaMaria Andriotis at AnnaMaria.Andriotis@dowjones.com