PHOENIX — Buoyed by an improving housing market, President Barack Obama on Tuesday proposed a broad overhaul of the nation’s mortgage finance system, including winding down government-backed Fannie Mae and Freddie Mac.

Obama outlined his proposals in Phoenix, the once foreclosure-riddled city at the epicenter of the nation’s housing crisis. The housing market in Phoenix, as well as in many other parts of the country, has rebounded robustly, with prices in the southwestern city up 66 percent from the low point in 2011.

Despite the nationwide gains, the president said sweeping housing reforms are still needed to ensure that a rejuvenated market doesn’t simply “re-inflate the housing bubble.”

“For too long, these companies were allowed to make big profits buying mortgages, knowing that if their bets went bad, taxpayers would be left holding the bag,” Obama told a crowd of more than 2,000 at an area high school.

While the president has previously endorsed overhauling Fannie and Freddie, his remarks Tuesday marked the first time he outlined his specific proposals for doing so.

The president wants to replace Fannie and Freddie with a system that would put the private sector, not the government, primarily at risk for loans.

The government would still be involved, both in oversight and as a last-resort loan guarantor.

Obama is also seeking guarantees that a private sector-led mortgage finance system would still ensure wide homeowner access to popular 30-year mortgages at fixed rates.

Obama’s mortgage reform priorities are largely in line with a Senate measure shepherded by Republican Sen. Bob Corker of Tennessee and Democratic Sen. Mark Warner of Virginia that would wind down Fannie Mae and Freddie Mac within five years.

Corker said Obama’s remarks were a sign of real and growing momentum behind efforts to shutter the mortgage giants.

Once flourishing, Fannie and Freddie were bailed out in 2008 by a $187 billion taxpayer-backed bailout. The two enterprises don’t make loans directly, but buy mortgages from lenders, package them as bonds, guarantee them against default and sell them to investors. The enterprises currently own or guarantee half of all U.S. mortgages and back nearly 90 percent of new ones.

The nationwide housing recovery has been providing critical support to the economy at a time when manufacturing and business investment have stagnated. Steady job growth and low mortgage rates in the past year have also fueled more home sales. The increased demand, along with a tight supply of homes for sale, has pushed home prices higher and encouraged builders to start more homes and create more construction jobs.