Graham said, "I've talked with the president his heart is right on this issue".
"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", he said.
Presidential historian Jon Meacham warned on Monday that immigration reform was being undermined by President Donald Trump's inability to shut out extremist members of his staff like policy adviser Stephen Miller.
"He's been an outlier for years", he said of Mr Miller.
White House aide Miller was formerly a staffer for Attorney General Jeff Sessions when he was Alabama's US senator - and has maintained a conservative stance on immigration in his new White Hour role, writing an immigration plan that includes wall funding and a crackdown on sanctuary cities. "He's been an outlier for years", said White House spokesman Hogan Gidley.
Graham was one of five Republican senators who broke party line to vote against the stop-gap funding measure that would have prevented a government shutdown Friday night.More news: Women too embarrassed to take cancer test
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Graham tweeted Sunday that some staff in the White House 'hold extreme and unrealistic views.
Although Republican senators and lawmakers don't usually name-check President Donald Trump's aides while criticizing them, the SC senator went a step ahead and slammed one specific key member of the commander-in-chief's xenophobic and bigoted entourage: Stephen Miller.
Graham on Sunday pointed to the meddling of White House aides, saying Trump initially wanted a cheaper proposal for the border wall following estimates that requested $18 billion for its construction. Before heading into a meeting with senators from both sides of the aisle, Graham laid into White House senior policy adviser Stephen Miller to a large assembly of reporters. "There's a deal to be had", Graham said.
Kelly believes Congress should pass legislation if it wants particular groups of immigrants protected from deportation - such as the people brought to the country illegally as children, called "Dreamers" by their advocates, who are at the center of the immigration dispute.
"Look, this is the president's agenda".
"The White House staff, I think, is making it very hard", he added.
In short, the Lindsey Graham proposal is amnesty now, border security maybe, illegal immigrants first, Americans last.