By Nick Timiraos 

A Senate committee approved a bipartisan measure to overhaul Fannie Mae and Freddie Mac on Thursday, a first step in what is shaping up to be a yearslong debate over revamping the nation's $10 trillion mortgage market.

The measure won support from 13 of 22 lawmakers on the Senate Banking Committee, with seven Republicans joining six Democrats.

While the bill is supported by the White House, analysts said Thursday's vote figures to be a hollow victory because it fell short of the larger majority needed to compel Senate Majority Leader Harry Reid (D., Nev.) to bring the bill up for a floor vote ahead of November's midterm elections.

The bill, sponsored by Sen. Tim Johnson (D., S.D.), the committee's chairman, and Sen. Mike Crapo of Idaho, the panel's top Republican, would construct a new market system in which private investors would take initial losses on mortgage securities that would carry government reinsurance.

A bipartisan consensus that the government needs to play a significant role to ensure the wide availability of the popular 30-year, fixed-rate mortgage has hardened over the past year. The overhaul proposal marked an rare instance in which leaders of both parties have worked together and with the White House on an issue that Washington had largely avoided in the years following the companies' 2008 taxpayer rescues.

(More to come.)

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