Sens. Dick Durbin, D-Illinois, and Lindsey Graham, R-South Carolina, speak during a news conference at the US Capitol September 5, 2017 in Washington.
Washington CNN  — 

Republican Sen. Lindsey Graham and Sen. Dick Durbin, a Democrat from Illinois, have teamed up to find a way forward on an immigration compromise to change the country’s asylum laws.

The bipartisan duo is working alongside the President’s son-in-law and senior White House aide Jared Kushner, who met with the lawmakers Tuesday.

The effort to overhaul the country’s asylum laws marks the latest breakthrough between Republicans and Democrats as they seek to remedy a growing crisis at the southern border. On Wednesday, the Senate Appropriations Committee advanced a bipartisan, $4.6 billion border supplemental bill to fund housing for unaccompanied minors and pay for transportation and other resources at the border.

Graham had unveiled his own plan to change the country’s asylum laws in May that would have required migrants to apply for asylum in their home countries or in Mexico. Graham’s plan also increased the amount of time migrant children be kept in custody from 20 days to 100 days and eased restrictions for officials to deport unaccompanied minors back to Central America.

Graham, the chairman of the Senate Judiciary Committee, had planned to move his bill out of committee Thursday, but told reporters Wednesday he’d delay the mark-up in an attempt to find common ground with Durbin, the Democratic whip.

“I feel we need to try,” Graham said. “(Durbin’s) asked to try and we’ll try.”

“We’re trying to find some common ground to deal with the problem I described,” Graham said. “What I want to do is lengthen the time to process a small child unit claim and to change the asylum laws so you have to apply in your home country or Mexico to stop the flow to the country.”

Graham and Durbin have worked for years on immigration compromises together. The senators were both part of the so-called Gang of Eight that crafted a bipartisan immigration bill in 2013 that passed the Senate but was never taken up in the House. They’ve also sought compromises to allow young people who entered the US illegally as kids the chance to stay in the US and earn citizenship.

“(Graham) had a chance to pass a Republican version of this bill on border policy, asylum policy, but I appealed to him and said that is going nowhere in the House, we need something that is bipartisan out of the Senate to really get this done in a bipartisan fashion,” Durbin said Wednesday.

Durbin said he and Graham will be working over the next “week or two to come up with bipartisan language.”

“I am hopeful, but I am not sure it is going to work,” Durbin said.

CNN’s Priscilla Alvarez contributed to this report.