Senate Democrats reintroduced a bill to expand refinancing for an estimated 13.5 million Fannie Mae and Freddie Mac mortgage borrowers.

Sens. Robert Menendez, D-N.J., and Barbara Boxer, D-Calif., will keep the origination cut-off date for the Home Affordable Refinance Program at June 2009 after a previous version expanded the program through June 2010.

"We made this change in a compromise with industry groups, and in response to issues raised by mortgage bond investors," Menendez said in a conference call with reporters Monday. "We've addressed every objection raised to the previous version."

The new version of the bill also eliminates previously proposed penalties for second lien holders and mortgage insurers who did not approve or transfer coverage in order to allow a HARP refinance to go through.

Both Menendez and Boxer expect to bring the bill to a vote either through committee or straight to the floor by the end of the month. Menendez said some Republicans on the Senate Banking Committee were attempting to attach "extraneous amendments."

"I've been in conversation with several colleagues that have shown an interest in the bill. We'll continue to court them," Menendez said. "It is my hope that we can do this in this month to move forward and take advantage of the low rates. Time is of the essence."

The new bill still allows a new servicer to avoid repurchase and warranty risk from the government-sponsored enterprises on the old loan. It also still prohibits the GSEs from charging any upfront fees to refinance a loan they guarantee, and it eliminates appraisal costs for all borrowers.

More than 1.5 million borrowers refinanced under HARP, since the program launched in 2009.

The Federal Housing Finance Agency eased guidelines last year to allow more deeply underwater borrowers to refinance, and the program spiked when lenders put in the new rules. Roughly 519,000 borrowers took advantage of HARP in 2012, already more than the 400,000 in all of last year, according to the FHFA.

The Obama administration renewed a push to get a new refinancing package through Congress. But despite some of the give by Menendez and Boxer Tuesday, enacting the bill remains a long shot in the Republican-controlled House.