Foreign envoys tell a story of watching Secretary of State Rex Tillerson with his head in his hands after reading a presidential tweet that undermined the very meeting he was in. It's a story they tell with some pity.

Amid rumors that the top US diplomat is frustrated and considering an early departure, friction between Tillerson, Trump and the White House team -- as well as headaches the President has created for his foreign policy officials -- are taking on new significance.

State Department officials have batted back reports of a "Rexit," with spokesman Heather Nauert calling them "false." A senior agency official who recently met with Tillerson told CNN that the secretary isn't letting any stress show, adding that, "he's been working with determination."

The secretary himself insisted today that he'll stay "as long as the President lets me." He added that his relationship with Trump is "good."

Lawmakers close to Tillerson say they think he'll persevere. "I think he's willing to deal with all the things that exist to try to ensure that this administration and our nation is successful in foreign policy," Senate Foreign Relations Chairman Bob Corker, a Tennessee Republican, told The Washington Post. Corker added that he doesn't "think Tillerson is on the verge of resigning."

But CNN's John King has been told that the former ExxonMobil CEO may be close to throwing in the towel. King reported Monday that Tillerson has a growing list of disagreements with the White House, that the top diplomat's friends say his frustration has been building over the past six months, and that Tillerson doubts the situation will improve.

Differences between Trump and Tillerson have been on display from the start -- as have the ways in which Trump has made Tillerson's path a bumpier one.

Tillerson has encountered White House interference with his staffing choices, he's had to contend with Trump adviser and son-in-law Jared Kushner stepping in on high-profile policy issues, including the Middle East, and the 65-year-old Texan has often had to deal with the undermining effect of presidential tweets that contradict his policy statements.

Tillerson has carefully acknowledged some of the challenges, telling an audience at the World Petroleum Council that his old job was a lot "easier" and noting the President's relative inexperience in global politics.

"In my old life, I spent a lot of time around the political world because I had to deal with governments all over the world so I'm quite, I'm quite comfortable in these settings," Tillerson said in early July. In contrast, he said, "We have a president that doesn't come from the political world."

Asked about Trump's disruptive tweets on foreign policy a month earlier in New Zealand, Tillerson had said that "the President has his own unique way of communicating with the American people and the world and it's served him pretty well."

Asked the same question in July, Tillerson sounded more subdued. He cited the "dynamic" situations he encountered as an oil executive. "Yes, things are going to happen," he said, speaking in Istanbul. Then he added, "it is more difficult, because of just the elements we talked about."

Mexican diplomats have had a front row seat to observe just how difficult Tillerson's job can be. In late February, Tillerson and Homeland Security Secretary John Kelly traveled to Mexico to repair damage done by Trump's dismissal at the launch of his presidential campaign of Mexicans as rapists, criminals and freeloaders, his declaration that the military would be used for deportations, and his insistence that Mexico would pay for construction of a southern border wall.

While they were meeting with the Mexican foreign minister, a Mexican diplomat tells CNN that Tillerson and Kelly got word of remarks Trump had made to manufacturing CEOs that "we're getting really bad dudes out of this country and at a rate that nobody's ever seen before and they're the bad ones and it's a military operation."

Tillerson and Kelly put their hands to their heads, the diplomat said, their frustration and disbelief visibly apparent as the meeting was abruptly stopped. The two sides had to separate, regroup and reconvene later.

In the months since, Trump hasn't stopped dropping presidential tweet bombs on his own officials.

Trump has contradicted his Cabinet members on NATO -- criticizing it while Tillerson and others reaffirmed US support for the alliance.

More recently, he disrupted Tillerson's work on Qatar, after other Gulf nations cut relations with Doha, accusing it of terrorism. While Tillerson worked to get all parties to resolve the issue, Trump sent out a 140-character blast that planted him firmly against Qatar and in the camp with Saudi Arabia, Bahrain and other nations.

And in June, Tillerson and Defense Secretary James Mattis said they'd made progress with Chinese officials on containing the nuclear threat from North Korea. Trump hit social media to say that attempts to work with China on Pyongyang had "not worked out."

Strains are showing in the President's relations with other Cabinet officials, most notably with his first and most public supporter, Attorney General Jeff Sessions. And some observers have pointed to friction between White House insiders like Steve Bannon and officials like Tillerson and the colleagues he works with closely, including Defense Secretary Jim Mattis, National Security Adviser H. R. McMaster, and Gen. Joseph Dunford Jr., the chairman of the Joint Chiefs of Staff.

"There is dissonance," Corker told The Washington Post . Referring to Tillerson, Mattis and McMaster, he said, "these three are focused on you know, longer-term outcomes" that can conflict with "a chalkboard in Bannon's office that lays out the campaign promises which were made."

"He's a prolific phone caller and receiver of phone calls. And so that input that he's getting sometimes conflicts with I would call the three principles," Corker said. "In addition to that, there are voices within the White House, sometimes as I understand it, they're the voices that he sees very first thing in the morning and so, you know, sometimes those can move things in a direction that are different, again, from the three principals that are out there, you know, working towards a different end. And so yes, I mean, it exists. We see it, it plays out in many ways publicly, sometimes in conflict with efforts that are underway and sometimes in ways that can, in fact, undermine what is occurring."

"The input that he's getting sometimes conflicts with I would call the three principles," Corker said. "It plays out in many ways publicly, sometimes in conflict with efforts that are underway and sometimes in ways that can, in fact, undermine what is occurring."

On Tuesday, back at work after a brief, unannounced vacation, Tillerson ignored questions about his reported frustration and emphasized his busy and productive schedule. He boasted of a "very good working breakfast this morning with Speaker [Paul] Ryan" as well as a "lengthy meeting with the Vice President at the White House on foreign policy issues today."

Smoke signals about Tillerson's unhappiness appeared with June stories that suggested he lost his cool at a White House meeting about staffing the State Department. Tillerson had told State Department employees that Trump gave him control over hiring, but the White House has nixed at least two of his picks for senior positions.

Even before Tillerson took the job, it became clear during his confirmation hearing that he disagreed with the President on a slew of policy issues: the Trans-Pacific Partnership, climate change, Russian interference in the election and its aggression in Ukraine, the idea of a registry for American Muslims, respect for Mexico, and the proposal that Japan and South Korea should consider developing nuclear weapons.

A more recent catalyst for rumors that the secretary of state is mulling an early exit are apparent divisions over the Iran nuclear deal, differences that led to arguments last week among the President, Tillerson and other Cabinet members.

Every 90 days, the White House must certify to Congress that Iran is complying with the deal signed in 2015. At the most recent mid-July review, Tillerson, along with the rest of Trump's senior national security team, agreed Iran was in line.

On July 17, Tillerson prepared to issue the announcement. Allies and lawmakers were told, a press briefing was announced -- and then the whole thing was put on hold.

At the 11th hour, Trump had objected. The President wanted to hear what would happen if he declared Iran non-compliant, an administration official told CNN.

The official said there were voices within the White House arguing that Iran wasn't complying, but the source heard no concrete examples of how Tehran was violating the deal. Tillerson argued that Iran was, in fact, in compliance and that the administration should proceed. It was an argument he had with Trump personally, but it took until the mid-afternoon before a decision was made to go ahead.

The State Department announcement was delayed until the following day.

This article has been corrected to more accurately reflect what Senator Corker told the Washington Post.