South Carolina Republican Sen. Lindsey Graham criticized White House staffer Stephen Miller on Sunday, the second day of the federal government shutdown, depicting him as an obstacle to reaching an immigration deal.

"As long as Steven Miller is in charge of negotiating immigration we are going nowhere," Graham told reporters. "He's been an outlier for years."

Miller, a key senior policy adviser to President Trump, is an immigration hardliner who helped scuttle a 2013 immigration compromise during his time as a staffer to then-Sen. Jeff Sessions.

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Graham is a key member of a bipartisan group of senators trying to reach a deal to resolve the shutdown and a member of one of the smaller bipartisan groups trying to find an acceptable compromise on DACA, the program that allowed undocumented immigrants brought to the U.S. as children to stay in the country. Senate Democrats are filibustering a short-term government-funding bill over the impasse over DACA.

"I talked with the president. His heart is right on this issue," Graham said of Mr. Trump's stance on immigration, but continued saying, "I think he's got a good understanding of what will sell and every time we have a proposal it is only yanked back by staff members."

His comments reflected frustration over his now-infamous meeting with Illinois Democratic Sen. Dick Durbin and the president two weeks ago about a potential immigration deal. Both senators were under the impression that the were going to have a private meeting with Mr. Trump, only to be joined by immigration hardliners including Sen. Tom Cotton and Rep. Bob Goodlatte, both of whom helped turn the President against the Graham/Durbin proposal.

Graham's criticism of the staffer is unusual because under ordinary circumstances members do not usually name-check White House aids. Indeed, Republican Senators often bristled when asked to react to controversial comments made by former Mr. Trump adviser Steve Bannon during his time at the White House.

On the other hand, Democrats have been less shy about calling out controversial White House staffers like Bannon. And Tennessee Republican Sen. Bob Corker, during the run up to his feud with Mr. Trump, also called out White House staffers.

On Sunday, Graham made clear that his criticism didn't extend to John Kelly, the White House chief of staff. He tweeted, "President Trump has expressed a desire to have border security with compassion on immigration. #winningcombination. General Kelly is tough but reasonable. Some other staff in the White House hold extreme and unrealistic views. They hold us back from getting a solution."

Graham then told reporters Kelly, "has brought order to chaos in the White House," joking, "I don't know what medal you could give him but he has really created a process at the White House that is far more orderly."