"Shutdown continues, no end in sight," Graham tweeted on Thursday afternoon.
While the President was at the border, Democrats in the Senate protested the refusal of GOP leaders to allow Senate votes on funding bills passed by the House, again blocking efforts to start work on a bipartisan foreign policy bill dealing with the Mideast.
"It really doesn't make any sense to deal with a government shutdown by shutting down the Senate," complained Sen. Marco Rubio (R-FL), who for the second time this week saw action on his bill held up by Democrats.
"People are being hurt, it's got to end, and it's got to end now," said Sen. Ben Cardin (D-MD) to a rally of federal workers on Capitol Hill, as they demanded that the Senate act on bills to re-open those parts of the federal government which were shuttered starting on December 22.
"We love our jobs," said Brian Ching, a NASA engineer who works at the Kennedy Space Center in Florida, as the crowd of union supporters chanted, "Back to work! Back to work!"
"Mr. President, end this shutdown now!" said one union leader to cheers.
The union rallies came a day before some 800,000 federal workers would miss their first paycheck because of the shutdown dispute, with no sign of any negotiations between the White House and Democrats, a day after President Trump stormed out of a meeting with Democratic leaders.
"I think there is a middle ground," said Sen. Rand Paul (R-KY), who urged both sides to simply split the difference on what the President wants for a wall, and what Democrats don't want to give, suggesting around $2.5 billion to reporters, though that number has been rejected by the President.
While President Trump was visiting the border in Texas, Vice President Mike Pence traveled to Capitol Hill to meet with reporters, saying no decision had been made on declaring a national emergency, in which the President would unilaterally tap other funds to build a wall or other barriers without the consent of Congress.
"I have an absolute right to declare a national emergency," the President told reporters at the White House before flying to Texas, indicating that if there's no deal, he would likely choose that option.
But while there is strong support for the President on the wall - Republicans have notably expressed public reservations about a national emergency decision by the President, concerned by the precedent it might set, a possible legal challenge, and what monies Mr. Trump might tap for wall construction, especially if it comes from the Pentagon.
"I’m opposed to using defense dollars for non-defense purposes," said Rep. Mac Thornberry (R-TX), the top Republican on the House Armed Services Committee.
"I adamantly opposed to the money coming from military construction," said Rep. Mike Turner (R-OH), worried - like other lawmakers - that any move by the President to tap those accounts could put on hold construction plans at local military bases in their districts.
Even some of the President's more vocal supporters agreed such a declaration would not be the best outcome for those in favor of a wall.
"I think there's some concern - I believe he's heard those - about how that power could be used by future Presidents for other reasons," said Sen. Rubio.
"I think we should do it legislatively," Rep. Jordan said on Thursday, "because if he (Trump) goes the emergency route, I'm convinced it's going to wind up in court."
In the halls of Congress, it was obvious that no deal was in the cards between Congress and the President, as the Senate left for the weekend by 2:30 pm on Thursday afternoon, while the House was passing funding bills which the White House said would be vetoed.