Graham, R-S.C., told CNN State of the Union host Dana Bash he believes a deal that would provide the $5 billion of funding for border security in the form of a wall or other barrier and a three-year one-time renewable work permit for about 70,000 people under the Deferred Action for Childhood Arrivals program "might save the day in the Senate."
"At the end of the day, there's a deal to be had," he said. "We need to start talking again."
He added that Democrats had previously rejected the deal and President Donald Trump "is not going to walk away from this fight" without funding for border security, adding he was "not asking anybody to do something new" but rather "rearrange the deal that was offered before."
"So to my Democratic friends, there will never be a deal without wall funding and many Republicans are going to offer something as an incentive to vote for wall funding that you have supported in the past," Graham said.
Graham also added he hoped "calmer heads will prevail."
White House chief of staff John Kelly, who is set to leave Trump's administration Wednesday, discussed the proposition of a border wall in an interview with the Los Angeles Times published Sunday, saying "to be honest, it's not a wall."
"The president still says 'wall' -- oftentimes frankly he'll say 'barrier' or 'fencing,' now he's tended toward steel slats," Kelly said. "But we left a solid concrete wall early on in the administration, when we asked people what they needed and where they needed it."
Trump Senior Adviser Kellyanne Conway responded to Kelly's comments while appearing on Fox News Sunday, referring to the distinction between a wall and other forms of border security as "a silly semantic argument."
"People who just want to say 'wall, wall, wall' want it to be a four-letter word and not respect what Customs and Border Patrol and ICE tell us they need, which is enhancements at the border," she said.
"There may be a wall at some places, there may be steel slats, there may be technological enhancements," Conway added. "But always saying 'wall' or 'no wall' is being very disingenuous and turning a complete blind eye to what is a crisis at the border when it comes to the drugs pouring in
Trump on Sunday tweeted referencing an Associated Press poll released Saturday showing support among veterans for his handling of border security and again blamed Democrats for the partial government shutdown as it entered its second week.
"Great work by my Administration over the holidays to save Coast Guard pay during this #SchumerShutdown. No thanks to the Democrats who left town and are not concerned about the safety and security of Americans!" he wrote, referencing Senate Democratic leader Charles Schumer.
He previously blamed Democrats for the deaths of two Guatemalan children while being held in the custody of U.S. officials by suggesting immigration policies he stated Democrats support were responsible for putting the children in harm's way.
Conway discussed Trump's comments on State of the Union Sunday, saying the United States "simply cannot absorb" groups of migrants attempting to enter the country as families or unaccompanied minors, noting that "very tragically, it does results in some deaths."
"We're allowing these people to take their children on a perilous journey without correcting the falsities. They're being promised something that people can simply not deliver upon," she said. "We don't want people to subject these children to these perilous journeys."