Home prices surged during the first quarter at their fastest pace in nearly seven years, the latest sign of a sustained warm-up in an economic recovery that has otherwise been marked by starts and stops.

The housing-market revival—and an accompanying report on consumer confidence—adds new grist for a debate inside the Federal Reserve about how far to push its easy-money policies, including an $85 billion-a-month bond-buying program which has helped to keep mortgage rates near historic lows, boosted asset prices and begun to stimulate hiring and spending.

Fed officials say they have been considering when to wind down the program. Signs of a stronger housing market could give confidence to officials who want to be sure that the economy can stand on its own, without the bond buying. Still, they have been reluctant to get too enthusiastic about signs of an upturn, in part because the economy has disappointed before.

The latest reports were big factors driving financial markets Tuesday. Stock investors, encouraged by the strong data, sent the Dow Jones Industrial Average to a new high. At the same time, yields on the benchmark 10-year Treasury note climbed to 2.132%, the highest level in 13 months. Bond yields rise as prices fall, and the gain could have been a response to strong data—and also the prospect of a less active Fed.

A sold sign is posted in front of a home in Phoenix. In March, prices were up by 22.5% in Phoenix from one year earlier. Getty Images

"The data we're seeing tells the Fed we're moving in the right direction, but I don't know if it's enough to really have the Fed scale back what they're doing," said Scott Buchta, head of fixed-income strategy at Brean Capital LLC.

The market reaction reflected rising optimism that the U.S. economy has shaken off a potential spring slowdown and is on track for steady gains. Home prices in March rose 10.2% from a year earlier, the largest annual gain since prices began to fall in 2006, according to the Standard & Poor's/Case-Shiller national index released Tuesday. Prices increased 1.2% in the first quarter, the first increase for the typically slower winter period since 2006. In a separate report, the Conference Board, a private research group, said Tuesday that its index of consumer confidence in May increased to the highest level since February 2008.

The pace of home-price gains has raised concerns among some economists over whether low mortgage rates have stimulated unsustainable home-price inflation—the proverbial bubble that some critics of Fed policies have feared. The average rate on the 30-year fixed mortgage, which is closely tied to yields on the 10-year Treasury, rose to 3.9% on Tuesday, the highest level in a year, according to HSH.com, a financial publisher.

At current interest rates, homes would still be affordable even if prices rose by 20% from their current levels, said John Burns, chief executive of a home-builder consulting firm in Irvine, Calif. "But what happens when [mortgage] rates go back to 7%? We're going to have a bust." He added, "If I were the Fed, I would stop buying mortgage-backed securities today."

The worries about rapid price growth look especially founded in more expensive markets such as San Francisco, Los Angeles and San Diego that have witnessed double-digit price gains over the past year. While home prices still look cheap on a historical basis, "the trouble is that that impression is almost entirely the function of low mortgage rates," said Stan Humphries, chief economist at Zillow Inc. Cheap credit is "distorting housing considerably," he said.

Others say it is too soon for alarm. Price gains largely reflect a rebound from low levels and prices remain largely in line with their long-run relationship between incomes and rents, said Christopher Thornberg, an economist with Beacon Economics in Los Angeles. "Could this thing go on too long? Absolutely," he said. "Could it turn into the next bubble? Absolutely. But we're not there yet, so I'm not going to start screaming 'Bubble.' "

The housing market has witnessed a surprising and sharp turnabout over the past year, first as investors have soaked up an oversupply of foreclosed properties and later as rising rents, low mortgage rates, and improving consumer confidence have unleashed pent-up demand for housing. Part of the rebound has been a function of the slump's depths. Home prices remain down by 28% from their 2006 peak and have returned to levels last seen in late 2003.

Top 10 Cities With Rising Home Prices Houses in Arizona. Justin Sullivan/Getty Images

Rising home prices should help buoy consumer confidence and the broader economy because houses represent the largest financial asset for many Americans. The home-price gains also should help housing markets by freeing more homeowners from being underwater, where they owe more than their homes are worth. Many would-be home buyers have complained in recent months over a paucity of desirable homes for sale, and home builders have taken advantage of supply constraints to boost prices. If price increases encourage more sellers to test the market, the new supply could boost sales volumes while tamping down price growth.

There are already signs that home prices are poised to stay hot this summer because the inventory of homes for sale is still depressed and buyer urgency has intensified. "There's not enough supply to satisfy demand, so we will continue to see upside surprises over the next few months," said Ivy Zelman, chief executive of Zelman & Associates, a housing research and advisory firm.

The Case-Shiller index can sometimes overstate the magnitude of price increases because it includes foreclosures. Over the past year, the share of foreclosed property sales has fallen, particularly in California cities, Las Vegas, and Phoenix, which have posted the largest year-on-year price growth. Because foreclosures and other distressed properties sell at a discount, prices tend to look worse as the share of foreclosed-property sales rise and they look better as the share drops. A separate index released Tuesday by Lender Processing Services Inc. showed that prices rose 7.6% in March from one year earlier.

Prices rose from one year earlier in all 20 cities measured by the Case-Shiller index, with the largest gains in the West or in markets pummeled by the housing bust. In March, prices were up 22.5% in Phoenix from one year earlier and 22.2% in San Francisco.

As demand outpaces supply, more real-estate agents are working to drum up leads by telling homeowners about intense buyer demand in their local neighborhoods. Jenel Lee, who owned a one-bedroom condominium in Seattle's Belltown neighborhood, received a postcard in the mail earlier this year from Sam Bryan, an agent representing a buyer that was ready to make an offer with a 50% down payment on a unit like hers.

"They're getting more aggressive letting homeowners know what's going on," says Ms. Lee, a 38-year old who works for a food importer. After renting the condo for the last four years, she drew six offers in less than one week after listing the unit in April. It closed for slightly more than the $275,000 asking price. She and her husband are planning to build a new five-bedroom home in a sought-after neighborhood a few minutes north.

Mr. Bryan, the real-estate agent, said he sent 200 postcards to unit owners like Ms. Lee in a bid to "flush out the interested sellers." The front of the postcard said in big letters, "Got Belltown Condo?"

Write to Nick Timiraos at nick.timiraos@wsj.com