"I have never been more depressed about moving forward than I am right now", Graham told reporters. "Somebody has got to get some energy to fix this".
This comes as the partial shutdown of U.S. government agencies stretches into a fourth week, as he has failed to secure the requested funding for the border wall in the House of Representatives.
"When, during the campaign, I would say Mexico is going to pay for it, obviously I never said this, and I never meant they're going to write out a check", Trump told reporters at the White House on Thursday before heading to the southern border. "And when I said "Mexico will pay for the wall" in front thousands and thousands of people, obviously they are not going to write a check".
The impasse between Congress and the White House appeared to hit another snag Wednesday when Trump walked out of a White House meeting with congressional leaders, calling the discussions a "total waste of time".
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"Pelosi's refusal to negotiate on funding for a border wall/barrier - even if the government were to be reopened - virtually ends the congressional path to funding for a border wall/barrier", Graham said in a statement.
Trump has reiterated his threat to declare a national emergency, which could give him the upper hand in his fight for border wall funding with Democratic lawmakers, who gained majority in the House of Representatives following the November midterm congressional election.
"If this doesn't work out, probably I will do it".
On Thursday, the president said he would declare a national emergency to compel funds for the wall if he couldn't make a deal with Democrats. Trump has said he plans to erect a steel wall over 30 feet tall along 200 miles of the US-Mexico border. "If this doesn't work out, I probably will do it, I would nearly say definitely", Mr. Trump told reporters, adding later, "If we don't make a deal, I would say 100 percent but I don't want to say 100 percent".