They also want better service and transparency.
Although most respondents to the survey showed satisfaction with primary banks (81 percent), many expressed frustration that could push them to find a lending alternative. Top on the list of gripes: "slow execution." More than half of consumers say the process needs to be faster. Others cited difficulty with communications, inability to track the status of the application and even untrustworthy advice.
"I think they trust good service, word of mouth and experience, and when they get those things, the survey shows, it's less about them being a bank and more about reaching the values that are core to them. That's trust, service and price," adds Hautop.
Familiarity is also a driving factor.
Most banking is done online today, with many consumers rarely if ever going into a retail bank. These same consumers are making weekly trips to discount retailers. As the mortgage market now shifts from a largely refinance trade to purchase applications for a home, banks need to take that level of exposure and the information about a captive audience and translate that into increasing market share.
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Warehouse giant Costco got that message last spring, announcing a new mortgage lending program with New Jersey-based First Choice Bank. Costco itself doesn't lend but gathers quotes from other lenders, so that the customer can comparison shop, essentially a marketing service.
Bottom line, consumers want to trust their lenders again. Retail giants don't have to deal with the legacy of lending ills uncovered after the housing and mortgage crash. Armed with new federal regulations designed to insure truth, clarity and transparency in lending, consumers now feel more protected when shopping for a mortgage and may be more willing to shop outside the bank.
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—By CNBC's Diana Olick; Follow Her on Twitter @Diana_Olick and on Facebook.
Questions? Comments? RealtyCheck@cnbc.com