Over the summer, Trump had a contentious phone call with Murkowski and also berated her on Twitter, accusing her of letting the country down for voting against GOP plans to gut Obamacare.

But behind the scenes, Trump has since tried to make nice, inviting her to a private lunch earlier this month -- something that seems to have defused tensions, at least for now. Murkowski had to decline the invite because she had a commitment in Alaska, but she said the gesture was "very kind."

"I don't think there's a sourness," Murkowski said of their relationship. "If there were, I don't think he would continue to be engaging -- and that's what we are going to continue to do."

The private outreach is part of a more concerted White House effort to repair relations Trump damaged with his own party over a tumultuous August, following the failure of the GOP effort to repeal Obamacare and a slew of White House controversies, including the President's controversial handling of a deadly protest in Charlottesville.

And it now comes at a critical time: The White House is desperately trying to revive its health care push before a September 30 deadline -- and needs senators like Murkowski to salvage the President's agenda. But of course, it hasn't always been smooth -- especially as Trump has tried to circumvent Republicans and cut deals with Democratic leaders -- over the GOP leadership's objections.

"I think it's like a thunderstorm," Sen. John Cornyn, the No. 2 Senate Republican, said of his conference's relationship with Trump. "After a while it clears up, the sun comes out and everything is OK."

The White House seems to recognize that bruised egos on the GOP side won't help Trump's efforts to revive an agenda stuck on Capitol Hill. And the President has won over some key Republicans by announcing he'd campaign with Sen. Luther Strange -- a favorite of the GOP leadership -- over a candidate in the Alabama GOP runoff backed by the party's Breitbart wing.

A White House spokesperson did not respond to a request for comment for this story.

Behind the scenes, Vice President Mike Pence has played a key outreach role, leaning on his longstanding relationships with senators -- even those who are frequent targets of Trump's derision -- to help underscore the point that they're all on the same team. He often praises senators for the job they're doing and says the President feels the same way, according to several senators who have spoken with the vice president.

After a rowdy Arizona rally where Trump derided Sen. Jeff Flake for being "weak" on borders and crime -- something he reiterated on Twitter the next day -- Flake got a phone call shortly afterward.

It was Pence on the line.

Flake, who said he has not spoken to the President since a tense face-to-face confrontation last year when Trump was a candidate, declined to divulge the contents of his conversation with Pence.

But he indicated it was a friendly chat.

"We're good friends," Flake said of Pence.

JUST WATCHED Flake: I'm not a 'rubber stamp' for Trump Replay More Videos ... MUST WATCH Flake: I'm not a 'rubber stamp' for Trump 02:07

Also in August, Trump appeared to equate white supremacists with counter-demonstrators after the Charlottesville violence -- something that prompted Tennessee Sen. Bob Corker to issue some of the toughest criticism of any Republican, questioning the President's "competence" and "stability" to be successful at the job. The comments sparked a sharp attack from White House spokeswoman Sarah Sanders and Trump himself tweeted that Tennessee is "not happy" with Corker.

But after those White House criticisms, Corker soon received a warm phone call -- from Pence. The message, according to sources briefed on the call, the President believes Corker is doing a good job as senator.

Corker declined to comment on the call.

But Corker said that he believed his relationship with Trump remained strong -- despite the dust-up. And Corker and Trump sat down face-to-face Friday for a private meeting, their first since the August flap, something that Corker's office said was "an extremely constructive" discussion that lasted more than an hour.

Corker told CNN Monday that the two men spoke "very briefly" about their August dustup, and then quickly moved onto a number of different issues.

"It was a very positive meeting," Corker said.

"No question," he said, "my relationship is just the same as it was before."

As a former congresman, Pence has long-standing relationships with a number of senators, plus he attends the weekly Senate Republican lunches, allowing him to constantly be in communication with GOP senators -- including ones who have sparred with Trump.

That includes Sen. John McCain, who has only spoken sparingly to Trump in recent months, including after the US strike against a Syrian airbase in April and after the senator revealed his cancer diagnosis this summer.

JUST WATCHED WH comments on Bob Corker, GOP relationships Replay More Videos ... MUST WATCH WH comments on Bob Corker, GOP relationships 01:04

Pence, on the other hand, has spent ample time talking with McCain, including in the Senate's ornate President's Room in the tense moments leading up to the Arizona Republican's decisive vote against a health care bill in July.

"He and I have known each other so long, he's traveled with me when he was in the House," McCain said of Pence. "It's not like there's outreach, we've known each other so long."

McCain added of Pence: "He's not a high-pressure guy, he tells us how great of a job we are doing."

A Pence aide noted that the vice president hails from the legislative world and is one of the main liaisons between the White House and the Hill, so he's often in contact with GOP senators. The aide added that Pence's conversations with senators, including McCain and Corker, have been policy focused.

Pence, too, has been in regular communication with Senate Majority Leader Mitch McConnell, who was on the receiving end of a Trump tirade over August, after the President repeatedly questioned McConnell's fitness for the job following the senator's remarks that Trump had "excessive expectations" about how quickly he could achieve his agenda in Congress. Things got so tense that Trump swore at McConnell in a contentious phone call on August 9 , sources said.

For the first time since their blowup, the two men sat down in early September for a private one-on-one meeting at the White House that lasted roughly 20 minutes, before they joined a larger group to discuss tax reform. And the meeting, sources said, was all business: discussing the party's agenda for the busy fall session. They didn't discuss their feud over August, but it was a professional discussion, the sources said.

Shortly after that meeting, McConnell would only say this when asked if the President's August attacks would undercut his agenda: "We're now in September, and we're confronted with the crises, and we intend to deal with them, this week."

Despite the often tense moments, many Republicans recognize the transactional nature of Trump's presidency, realizing full well that sometimes they are on good terms with him, sometimes they are not.

Asked if he finds that kind of relationship acceptable, Cornyn said bluntly: "What choice do I have?"