Trump's wall is winning adherents as support increased over the past year.
"Before he pulls the plug on the legislative option - and I think we are nearly there - I would urge them 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", Mr Graham said on Fox News Sunday.
"If we can't (get a deal) at the end of three weeks, all bets are off", Graham told "Fox News Sunday". See if we can get a deal.
Sen. Chris Coons, D-Del., called Graham's idea for a brief reopening of the government a "great place to start".
Congressional Democrats are engaged in a more than three-week battle with Senate Republicans and the White House over funding for a wall along the southern border. "But while we're having that debate, let's reopen the government".
Congress returns on Monday.
Trump continued making his case for the wall in Sunday morning tweets, arguing that it "will bring down the crime rate throughout the entire Country!"
The government shutdown is now the longest in American history, with no apparent end in sight. Roughly 800,000 government workers missed a paycheck on Friday. Functions not happening include some routine food inspections, timely release of market-moving agriculture data, and USA reviews needed for initial public offerings of stocks.
Democrats, meanwhile, seem determined at all costs to prevent the president from getting the wall he has often promised in his campaign-style rallies.
Last week, the talks broke down during a contentious meeting at the White House at which Trump walked out when told by House Speaker Nancy Pelosi, that she would continue to oppose the wall in border security negotiations, even if the government was reopened.
"So until that changes, there's not much left except the national emergency approach", Graham said on Fox.
Turning to Twitter on January 13 Trump again pushed for the wall and pointed to Democrats to join in on the talks. "I do have a plan on the Shutdown". But the Mexican government has refused and Trump is now demanding that Congress provide funding.
Sen. Richard Durbin of IL, the No. 2 Democrat in the Senate, said on NBC News' "Meet the Press" that centrist Senate Republicans who sought to broker a deal last week should step forward and make an appeal to their party's leader in the chamber, Senate Majority Leader Mitch McConnell of Kentucky. Almost 42 percent of subjects say they support a wall, which is up from 34 percent in January 2018. Separately, a CNN poll showed 56 percent of Americans oppose construction of Trump's hoped-for border wall.
"There's a real risk for Democrats if they go out and they do something to help President Trump, that their own base is going to go after them", Gingrich concluded. "The easiest solution to the shutdown, just give President Trump the money", he said. "As a candidate, this is what he talked about, and if there's any mandate he can claim from his election, it was better border security and keeping this nation safe". "Clearly the president's got authority under the law, but he's said he doesn't want it to come to that". Nine of 15 cabinet-level departments have been closed since December 22, after Trump refused to sign a stopgap spending bill that didn't satisfy his demand for at least $5 billion to fund a wall along the southwest border.
The shutdown became the longest on record at midnight Friday (0500 GMT Saturday), when it overtook a 21-day stretch in 1995-1996 under president Bill Clinton.
They are classed as essential employees and required to work, but many are calling in sick to protest the situation.
Miami International Airport on Saturday closed its least-used concourse due to a shortage of TSA officers, according to the Miami Herald. About a third of those surveyed - 32 percent - thought Democrats were more at fault, while 9 percent blamed each side equally. Some staff at the State and Homeland Security departments are also working without compensation. Senate Democrats, whose votes would be needed to advance such legislation, have not indicated whether they would back it because they want the government to reopen instead.