Democrats and the White House must drop the “blame game” and restart talks to reopen the government fully, weary lawmakers said Sunday as Capitol Hill lurched toward the new year without any sign that they will break the impasse over President Trump’s border wall.

Senior Republicans said an extension of legal status for “Dreamers” brought to the U.S. illegally as children could be the sweetener Democrats need to approve the $5 billion Mr. Trump wants for the wall and end the federal shutdown, which has entered its second week.

“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,” Sen. Lindsey Graham, South Carolina Republican, told CNN’s “State of the Union.”

SEE ALSO: Sen. Richard Shelby decries ‘blame game’ over shutdown: ‘We all lose, and we all kind of look silly’

Democrats, who can block bills in the Senate because of the 60-vote requirement even before they take control of the House later this week, said they are willing to negotiate a deal but remain leery, arguing that Mr. Trump will move the goal posts if they compromise with Republicans.

For now, Democrats seem content to wait until they retake the House, giving them a chance to pass a bill on their terms and dare the Republican-led Senate to remain in a shutdown.

While the warring sides talked past each other, federal workers caught in the crosshairs mulled side gigs to make ends meet during the shutdown. Their next government paycheck is nowhere in sight, and the fight over border security turned even nastier over the weekend.

SEE ALSO: GOP Rep. Carlos Curbelo urges Trump to look to McConnell to reopen the government

Mr. Trump blamed Democrats for the deaths of two migrant children at the border.

“Any deaths of children or others at the Border are strictly the fault of the Democrats and their pathetic immigration policies that allow people to make the long trek thinking they can enter our country illegally,” he tweeted Saturday. “They can’t. If we had a Wall, they wouldn’t even try!”

White House counselor Kellyanne Conway swatted aside critics who said the tweet offered little sympathy for the Guatemalan children and their families.

“I think the president’s point is an important one,” Mrs. Conway told CNN. “He stayed in Washington to negotiate border security.”

She said Mr. Trump already dropped his funding request for the wall to $2.5 billion and was waiting for Democrats to offer money for a deal.

Democrats said there are better ways to secure the border than to build a wall.

“We do need comprehensive immigration reform. In the end, we need to know what the president wants, and hopefully he’ll stick to it,” Sen. Jon Tester, Montana Democrat, told CBS’s “Face the Nation.” “We need some predictability.”

Sen. Richard C. Shelby, Alabama Republican and chairman of the Appropriations Committee, appeared to fault all sides for the impasse. He said the lack of negotiations made everyone in Washington losers and that it was time for Mr. Trump to stop tweeting and for Democrats to show their cards.

“Nobody’s going to win this type of game,” Mr. Shelby told CBS’s “Face the Nation.” “We all lose, and we all kind of look silly.”

The latest shutdown, which began Dec. 22, has been less painful than previous ones. Some of it went unnoticed over the initial weekend and Christmas holiday.

Yet the biggest factor is that Congress managed to fund the Defense, Health and Human Services, Labor, Education and Veterans Affairs departments in September. Those departments account for some 75 percent of annual discretionary spending. As a result, troops and many federal departments are still on the job and getting paid.

The fight revolves around Mr. Trump’s pledge to build a wall along the southern border. The proposed barrier was a part of his campaign and, after bruising Republican midterm losses in Congress, the president sees the standoff as his last reasonable chance to secure ample funding.

Democrats say they won’t budge from their opposition. They argue that a wall is ineffective and sends the wrong message to the world. Citing Mr. Trump’s own words on the campaign trail, they said Mexico was supposed to pay for it anyway.

The White House of late has said it will settle for fencing in the form of steel slats, noting that Democrats including Senate Minority Leader Charles E. Schumer of New York have supported fencing in the past.

Mr. Graham said if Democrats supported fencing in the past, then there is no reason — besides politics — to dig in now.

“The bottom line is, they want Trump to lose more than they want the country to win, I fear,” Mr. Graham told CNN. “At the end of the day, there’s a deal to be had. But everybody’s changing their position here. And most Americans are pretty tired of it.”