WASHINGTON — In another sign that the housing market has strengthened, President Barack Obama on Tuesday outlined his long-awaited ideas for overhauling mortgage finance giants Fannie Mae and Freddie Mac to significantly reduce the government’s risk in any future credit crisis.

In an appearance in Phoenix, Obama endorsed bipartisan efforts in the Senate to wind down the two companies and end their longtime implicit guarantee of a federal government bailout. That dread prospect, once thought improbable, was realized in the fall 2008 financial crisis; Fannie Mae and Freddie Mac, then bankrupt, were rescued by the government at great cost to taxpayers, who only now are being repaid.

The president made clear that he would only sign into law a measure that puts private investors primarily at risk for the two companies, which buy and guarantee many mortgages from banks to provide a continuing stream of money for lenders to provide to additional homebuyers.

An acceptable measure also must specify the government’s role and liabilities for Fannie Mae and Freddie Mac, and — unlike legislation in the Republican-controlled House — must ensure Americans’ continued access to a 30-year mortgage at a fixed interest rate.

After years in which the formerly formidable Fannie Mae and Freddie Mac and their congressional allies blocked proposals for payment of fees or risk premiums, Obama is calling for financial institutions to pay an assessment to the government on the value of mortgage-backed securities. Under his proposals, the revenue would help finance assistance for borrowers and subsidize construction of homes and rental properties that would be affordable to lower-income Americans.