Texas Republican Rep. Jeb Hensarling told a conference of mortgage bankers Wednesday his 2013 bill that would have ended Fannie Mae and Freddie Mac still remains the best path forward for reform of the housing finance market.

Appearing at a meeting of the Mortgage Bankers Association Wednesday morning in downtown Washington, the conservative chairman of the House Financial Services Committee said the legislation "still represents the best vehicle for reform."

The bill cleared the committee with Republican votes in 2013, but failed to advance in the House thanks to finance industry skepticism.

Notably, it would have dissolved the bailed-out mortgage giants Fannie Mae and Freddie Mac. In their stead, it would have created a privately-run utility to facilitate the creation of mortgage-backed securities and support a secondary market for home loans.

The bill would have removed government backing for mortgage-backed utilities, though. At the time, many industry groups, including the Mortgage Bankers Association, argued eliminating the government backstop for mortgage-backed securities would make 30-year fixed-rate loans unavailable.

Hensarling on Wednesday called it a "fable" that his legislation would have ended the 30-year fixed-rate mortgage.

The lawmaker has been meeting with Senate Banking Committee Chairman Mike Crapo of Idaho and Treasury Secretary Steven Mnuchin to discuss housing finance reform. All parties have said they aim to address the status of Fannie and Freddie later this year or early next.

"I'm confident this is the Congress where we will finally achieve that goal," Hensarling said Wednesday.

He also said that he was reviewing outside proposals for overhauling the system of housing finance, including the one published by the Mortgage Bankers Association earlier this year.