"I have never been more depressed about moving forward than right now", said Graham Thursday.
On Saturday, the shutdown ran into a 22nd day, making it the longest government shutdown in United States history.
"Elections have consequences!" he declared, meaning the 2016 election in which "I promised safety and security" and, as part of that, a border wall.
Graham said he thinks Trump is willing to accept the $5.7 billion he has insisted on for the wall, along with some immigration measures Democrats might find acceptable, such as helping immigrants who were illegally brought to the U.S.as children.
"Before he pulls the plug on the legislative option, and I think we're nearly there, I would urge him to open up the government for a short period of time, like three weeks, before he pulls the plug, see if we can get a deal", said Graham, a South Carolina Republican.
On Sunday, Graham told Fox New's Wallace that the administration wants to make a deal but Democrats refuse the any deal that includes the wall.
The previous record dates back to the Clinton administration when a 21-day shutdown resulted from a clash between President Bill Clinton and the GOP Congress that lasted from December 1995 to January 1996.
Negotiations have been stalled again because the House adjourned and several Democrats headed to Puerto Rico for a vacation with lobbyists, much to the chagrin of President Trump who is waiting to work out a deal at the White House.
"More border security - let's have at it", Senator Mark Warner, Democrat of Virginia, said on CNN's "State of the Union".
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Many furloughed federal employees have taken to the streets to protest their situation. But the numbers also tell a story of a deepened partisan divide that has likely hardened due to the media attention on what has caused the shutdown in the first place: Trump's farcical wall.
The government shutdown is now well into its third week, making it the longest shutdown in history. Instead, McConnell last week vowed to stand with Trump in the wall debate, after the president met with all Senate Republicans and pleaded for party unity behind his wall quest.
Former White House communications director Anthony Scaramucci told the West Block's Mercedes Stephenson that the three-week shutdown is a grudge match symbolizing the bigger ideological fight between Republicans and Democrats, but warned the country will face serious economic harm if it doesn't end soon.
But while that might end the standoff and allow Congress to move to other priorities, some Republicans believe such a declaration would usurp congressional power and could lead future Democratic presidents to make similar moves to advance liberal priorities. Democrats have rejected his request and say they will not negotiate further until the government is reopened.
This week, the president is expected to sign legislation passed by Congress to provide back pay for around 800,000 federal workers who are not being paid during the shutdown.
As the shutdown drags on, President Donald Trump and congressional Democrats have not made progress toward any kind of agreement that would put an end to it.
But the Post poll also found a hardening of Republican support for a wall, with 70 per cent now saying they strongly support the wall, up from 58 per cent a year earlier.
Funding for the border wall remains the sticking point in passing budgets to reopen the government.