Graham told his colleagues that bipartisan talks had already been underway with the White House, which have included Sen. Richard Durbin (D-IL), the second-ranking Democrat in the Senate.
"I am willing to deal with DACA," Graham said, addressing an issue which Democrats say would have to be part of any deal.
"I am not willing to continue the practice that our laws are generating, which is to tell everyone in Central America, the door is open," Graham added.
Both sides know that any compromise may include items which are anathema to many of their own supporters; one example would be a legal status for those covered under DACA and so-called "Dreamers," as Democrats want them to have a chance at a 10-to-12 year pathway to citizenship.
For many Republicans, that type of plan spells only one word: Amnesty.
Just over a month ago, Graham floated a bill which included asylum system changes, and urged President Trump to get on board with an immigration compromise.
"To get what you want, you've got to give something," Graham said in mid-May.
Graham has tried repeatedly in public to make the case to fellow Republicans and the White House that an immigration deal is in their best interest - otherwise nothing will get done in terms of legislative changes, and the number of people flowing to the border will continue to grow.
At a hearing last week, the acting DHS Secretary said Graham's 'Gang of 8' immigration bill from 2013 - which drew furious opposition in conservative circles and on talk radio, and was never voted on by the GOP House - would have certainly helped prevent the current situation at the border.
"We would have a very different situation at the border," said Acting DHS chief Kevin McAleenan, as he noted the plan would have bolstered the number of agents for the Border Patrol, ICE, and included other immigration enforcement improvements, such as up to 700 miles of new pedestrian fencing along the border, and high tech sensors.
But the "Gang of 8" bill foundered among House Republicans in 2013 because of the provisions dealing with the DREAM Act, and a pathway to citizenship for some of those already in the U.S. illegally.
And it was quickly obvious in the halls of Congress that the same type of concerns could doom any new effort to strike a deal in 2019.
The 2013 "Gang of 8" bill would have granted immigrants a provisional legal status in the U.S. for six years, and renewable for another six years, with a $500 fee.
After ten years, immigrants could then apply for a green card, and permanent resident status, as they would not be allowed to jump straight to citizenship.
Before any of those immigrants could even start getting a new legal status, the bill required that border security and fencing plans be in place first.
But that wasn't enough for Republicans - one reason many labeled the South Carolina Republican, Lindsey "Grahamnesty" for his efforts.
The Gang of 8 bill would also have ended the Diversity Visa Lottery Program - a frequent target of President Trump.