President Donald Trump and Senate Majority Leader Mitch McConnell are set for an interesting meeting at the White House on Monday.

Trump blames the Kentucky Republican for the health overhaul failure. Trump hints at tantalizing deals with Democrats and watches his former strategist work from outside the administration to bulldoze the Republican establishment on Capitol Hill.

There will be a chill in the air when Trump, a public official since January, and McConnell, first elected to Congress in 1984, sit down for lunch.

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"Mitch McConnell's not our problem. Our problem is that we promised to repeal and replace Obamacare, and we failed. We promised to cut taxes and we have yet to do it," said Sen. Lindsey Graham, R-South Carolina on CBS' "Face the Nation" on Sunday.

He added, "If we're successful, Mitch McConnell's fine. If we're not, we're all in trouble. We lose our majority and I think President Trump will not get re-elected."

Steve Bannon, back at Breitbart News after helping Trump win the presidency and serving in the West Wing, is committed to dumping the Kentucky Republican. In a speech to religious conservatives Saturday, Bannon put on notice some of those incumbents who are at risk of a challenge from his flank of the party. He said the lawmakers possibly can avoid that wrath if they disavow McConnell and meet other conditions.

"This is our war," Bannon said. "The establishment started it. ...You all are gonna finish it."

But according to someone familiar with the call, the president had phoned McConnell, hours after the Bannon speech, to discuss judicial nominations, next week's budget vote, and tax reform, showing the president and the majority leader are in regular contact about getting things done, CBS News' John Dickerson reports.

Meanwhile, Sen. Susan Collins, a Maine moderate who just passed up a run for governor and was a pivotal "no" vote on health care, said Bannon's rhetoric is exactly what the American people are tired of. "They don't want this hyper-partisanship. They want us to work together. And they want us to get things done," she said.

Collins, who's served in the Senate since 1997, added that Bannon's "over-the-top rhetoric is not helpful. Mitch McConnell is the Senate majority leader. The president needs him. I'm glad they're working together on tax reform and a lot of other issues. And I'm glad they're meeting this week."

McConnell responded to Trump's Twitter barrage after the failed health care effort by saying that the challenges of governing should come as no surprise.

"A lot of people look at all that and find it frustrating, messy. Well, welcome to the democratic process. That's the way it is in our country," McConnell said at a GOP event in Kentucky this summer.

Trump, a former Democrat himself, cut a deal with Democratic leaders on raising the U.S. borrowing limit and keeping the government running into the winter. The president has also talked about future arrangements, though his recent list of immigration demands soured Democrats who had seen an earlier opening for legislative progress.

Hard-right conservatives frustrated by the stalled agenda in Congress wrote in a letter last week during the Senate's break that McConnell and his leadership team should step aside. The senators' weeklong recess also drew criticism from the White House: "They're on another vacation right now. I think that we would all be a lot better off if the Senate would stop taking vacations, and start staying here until we actually get some real things accomplished," Trump spokeswoman Sarah Huckabee Sanders had said.

Meanwhile, a McConnell-backed political committee spent millions to support Alabama Sen. Luther Strange in a recent primary election and Trump endorsed him, but Bannon-backed Roy Moore prevailed. Moore, a former judge, has defied federal court orders, described Islam as a false religion and called homosexuality evil.

Senate Republicans had been upbeat about adding to their 52-48 edge in the chamber, especially with Democrats defending more seats next year — 10 in states Trump won in last year's presidential election. But the Bannon challenge could cost them, leaving incumbents on the losing end in primaries or GOP candidates roughed up for the general election.

"If we don't cut taxes and we don't eventually repeal and replace Obamacare, then we're going to lose across the board in the House in 2018. And all of my colleagues running in primaries in 2018 will probably get beat. It will be the end of Mitch McConnell as we know it. So this is a symptom of a greater problem," Graham said.

He added that Bannon "can't beat us if we're successful. And if we're not successful, it doesn't matter who tries to beat us, they'll be successful."