Updated 8:09 p.m. ET, 7/15/2014
(CNN) - Senate Republicans expressed growing confidence Tuesday that Congress will approve bipartisan legislation aimed at resolving a what's being called a "humanitarian crisis" resulting from the surge of child immigrants from Central America at the southern border.
"It is actually possible to come behind a bipartisan agreement to deal with a big problem," an upbeat Senate Republican Leader Mitch McConnell said about the proposal this week by Republican Sen. John Cornyn and Democratic Rep. Henry Cuellar, both of Texas.
Supporters say their plan would lead to the quick return of most young illegal migrants. That would come from eliminating part of a 2008 law that requires a hearing for all unaccompanied minors not from Mexico or Canada before they can be deported.
That time consuming hearing process is seen by many as a root problem of the current backlog at the border where tens of thousands of young illegal immigrants have crossed into the United States this year in a surge the Obama administration said was unexpected.
Congressional Republicans, however, have made clear they will not consider a pending $3.7 billion funding request from President Barack Obama to deal with the border crisis unless it includes policy changes that will deter similar immigration in the future.
"What our legislation would do is fill in the gap left by the President's request for more money," Cornyn said. "He's asked for $3.7 billion but not anything that would fix or solve the problem. We supply that."
The proposal includes expedited hearings for just some migrants. That concerns several senior Democrats who are influential on immigration issues and want to ensure all the children who come to the United States get due process under the law.
"You can't in 72 hours go ahead and make the case that your father got murdered in front of you. You can't make the case that the gang said join us or die if you don't have the time to produce documents, affidavits, certificates and what not. And so it is unacceptable to me to basically have a deal that undermines all those rights," said Sen. Robert Menendez, R-New Jersey who called the proposal "not humane."
Senate Majority Leader Harry Reid said he personally opposes the proposal.
He argued the law does not need to be changed to deal with influx of children and said the extra money the White House wants will resolve the crisis.
However, he didn't rule out bringing some legislation to the floor that would modify the Bush-era law in question.
Privately, Democrats acknowledged their caucus is divided. About half want to maintain existing legal rights and the other half is anxious to drop the fight and quickly approve the funding.
Senate GOP aides said they anticipated the Cornyn/Cuellar bill and separate legislation addressing related emergency spending would be taken up by the House late next week.
The Senate would then have to decide how to deal with the politically-sensitive measures just days before a five-week recess.
"They will fold like a cheap suit eventually," quipped Sen. Lindsey Graham, a South Carolina Republican. "Because the public is with us."
House Republicans have some of their own ideas that would be cheaper than Obama's request.
Texas GOP Rep. Kay Granger and other members of a working group set up by Speaker John Boehner outlined a set of policy changes at a meeting with Republican rank and file on Tuesday.
The working group wants to deploy the National Guard to the border. It also suggests bringing on retired judges to expand the now overloaded legal system and accelerate the process to return children to their home countries. It also proposes to add new enforcement powers for border agencies.
Arizona GOP Rep Matt Salmon, a member of the working group that visited Honduras and Guatemala over the weekend, said there's no common ground with the White House.
"The priorities that we're talking about – repatriating those children as quickly as possible – and where the President's is at – right now we're on different planets," Salmon said.
House Appropriations Committee Chairman Hal Rogers said last week the administration wants "too much money" and noted that some of the funding to address the border situation is already included in annual spending bills, including the homeland security spending bill the House passed last month.