WASHINGTON (CNN) -

Bipartisan legislation to curb the flood of unaccompanied minors at the southern border was unveiled Monday by two lawmakers from Texas.

Sponsors believe the compromise could lead to a quick exodus for most young migrants from Central America, as demanded by Republicans, while offering an appeals process for those who may be eligible to stay, which is supported by many Democrats.

It remains to be seen whether the proposal, authored by Republican Sen. John Cornyn and Democratic Rep. Henry Cuellar, can bridge the gap between the parties on the politically-volatile immigration issue.

But with Congress just three weeks away from its summer recess, it is likely to be prominently in the mix of solutions to the burgeoning crisis.

"My hope is that as people begin to see this is a bipartisan solution to a real problem they'll come on board," Cornyn told reporters.

The legislation would modify a 2008 law aimed at combating human trafficking by removing a requirement that all unaccompanied minors not from Mexico or Canada get a mandatory hearing before an immigration judge before being deported.

It is widely believed the hearing requirement in the Bush-era law contributed to the current problem by creating a backlog of cases.

Many Central American parents seem to believe their children will be able to take advantage of the lengthy process and ultimately stay in the United States.

The proposal, however, would allow children who are the victims of human trafficking or have a credible threat of persecution to appear before a judge to seek asylum.

The bill would require immigration judges to make a decision within 72 hours of a child making a claim.

"Most importantly to me is it actually gets the kids in front of a judge who can determine if there is any basis for relief under existing law," Cornyn said. "If there is not, than they would be repatriated."

Republicans who control the House have not decided if they will move the Cornyn/Cuellar bill. They are waiting to hear recommendations on Tuesday from a working group of lawmakers, which traveled this past weekend to Guatemala.

Last week, House Democratic Leader Nancy Pelosi did not reject the idea of treating all immigrant children the same, but she has not commented on the current proposal.

Homeland Security Secretary Jeh Johnson is scheduled to meet with a group of conservative southern Democratic House members on the issue on Monday.

Right now, Cornyn and Cuellar have no Democratic sponsors in the Democratic-led Senate. In fact, several influential Democrats -- including Assistant Majority Leader Dick Durbin of Illinois, Judiciary Committee Chairman Patrick Leahy of Vermont, and Foreign Relations Committee Chairman Robert Menendez of New Jersey -- have been highly skeptical of the emerging proposal.

Durbin said last week he could not support changing the 2008 law unless it met two criteria: That children appearing at immigration hearings have legal representation and that the United States have assurances about the quality of the "social infrastructure" those returned would face in their home countries.

"I want to make sure that the children who are going through this have proper advocacy and representation. In downstate Illinois, you wouldn't turn a 10-year-old loose in a courtroom without somebody standing next to them who can explain what is happening to them. I think that we owe to these children as well," Durbin said.

The Cornyn proposal relies on existing law that calls on the Department of Health and Human Services to seek out pro bono legal counsel but it does not require that immigrants have legal counsel.

He dismissed Durbin's proposal as "lawyering up."

"I think that would be a disaster for our immigration system. Basically saying everybody who enters the country illegally is entitled to a taxpayer paid-for lawyer. And you not only have a right to a lawyer at the immigration judge stage but you also have a right to appellate review. I mean these can obviously go on for years and that would serve the goal of some of the people who want to come because it would be years before they were ever deported," Cornyn said.