House Speaker John A. Boehner (R-Ohio)  (Evan Vucci/AP)

House Republicans are modifying their plan to address the record number of migrants the U.S.-Mexico border, with a framework to spend less than $1 billion gaining traction Friday.

"It's quite possible," said Rep. Kay Granger (Tex.), who  led a GOP working group on the border, when asked whether the cost of the package would be below $1 billion.

As they left a House Republican conference meeting, senior House GOP aides concurred, citing concerns from conservatives over the size and scope of legislation.

This latest development underscores the challenge facing House Speaker John A. Boehner (R-Ohio) as he works to find consensus on a border plan before the House adjourns Aug. 1 for a five-week recess, and comes after House GOP leaders unveiled a $1.5 proposal Wednesday, far less than President Obama's $3.7 billion request.

But by lowering the cost and concentrating the bill on more quickly processing the young children and families who have illegally entered the country, boosting funding for Border Patrol and tweaking a 2008 anti-trafficking law to make it easier to deport minors from Central America, House leaders hope to be in a position next week to win passage.

Republicans are down to "bare bones suggestions," Granger said.

Some House Republicans, however, remain unsure of whether the leadership will be able to win a majority of Republican support, in spite of Boehner's willingness to adjust his initial outline. Few also believe that House Republicans will be able to reach agreement with Senate Democrats on a final deal in the coming days.

Addressing President Obama's "Deferred Action for Childhood Arrivals" policy, which enables some young migrants to not immediately be deported, is also likely to not be part of the House GOP's plan and another reason some conservatives may oppose it.

"We've got to call on the president to cease those activities," said Rep. Steve King (R-Iowa), a longtime critic of bipartisan efforts to pass comprehensive immigration reform.

"John is going to lose 60 Republicans no matter what he does," said Sen. Lindsey Graham (R-S.C.), a Boehner ally. "Sixty Republicans are ready to vote against anything, so he may have to find some Democrats to vote with him."

But by paring down the plan, Boehner appears, for the moment, to have calmed his ranks, particularly to his right, and looks poised to get most Republicans to back him on the floor. Over the weekend, members will receive updates on the final version of the plan, aides said.

"It looks good," said Rep. Paul Ryan (R-Wis.), an influential House conservative, as he left the GOP meeting. Added Rep. Steve Scalise (R-La.), the incoming House majority whip: "The vast majority of our members want to solve this, do it in a targeted way that actually addresses the problem."

"It's all about getting to 218 votes," said tea party favorite Rep. Raul Labrador (R-Idaho), referencing the number of votes a bill needs to pass the lower chamber. "What we're hearing is good."

Senate Democrats are working on their own proposal, which would spend $2.7 billion to provide more resources at the southern border. Unlike the House proposal, that plan would not seek changes to the 2008 anti-trafficking law. Progressive activists oppose changes, but a handful of centrist Democrats, such Sen. Joe Manchin (W.Va.), are open to discussing adjustments to the law.