Tillerson is also taking time off at a time when the US faces multiple foreign policy challenges.
Nauert dodged a question about Tillerson's feelings over the White House's involvement in foreign policy decisions.
"The Secretary, as do all other Cabinet officials, meets with the President and the President's national security advisers and Cabinet members. That is something that's normal, that's customary. They sit down, they have a healthy dialogue and conversations about the heaviest and the weightiest foreign policy issues," she said.
Nauert added that Tillerson is not considering resigning and any reports saying so are "false."
Tillerson has "been very clear he intends to stay here at the State Department. We have a lot of work that is left to be done ahead of us. He recognizes that. He's deeply engaged in that work," she said.
He "does, however, serve at the pleasure of the President, just as any Cabinet official would," she added.
Tillerson has been on vacation this week -- Nauert said it had been planned for a while and is not related to reports of his dissatisfaction. Defense Secretary James Mattis is also currently taking time off.
Tillerson is returning to work on Wednesday and will meet with the Lebanese Prime Minister and Qatari Foreign Minister.
Nauert said that Tillerson has "the ability to go away for a few days on his own. Just taking a little time off. He's got a lot of work. He just came back from that mega trip overseas, as you all well know. Many of you were there for the G-20, and his other travel as well. So he's entitled to take a few days himself."
CNN's John King reported on Monday that Tillerson has a growing list of differences
with the White House, including a new debate over Iran policy and personnel. His frustration is hardly a secret and it has spilled out publicly at times
But friends sense a change of late.
For weeks, conversations with Tillerson's friends outside of Washington have left the impression that he, despite his frustrations, was determined to stay on the job at least through the end of the year.
But two sources who spoke to CNN on condition of anonymity over the weekend said they would not be surprised if there was a "Rexit" from Foggy Bottom sooner that that.
Both of these sources are familiar with Tillerson's conversations with friends outside Washington. Both said there was a noticeable increase in the secretary's frustration and his doubts that the tug-of-war with the White House would subside anytime soon. They also acknowledged he could have been venting after a tough week, a suggestion several DC-based sources made when asked if they saw evidence Tillerson was looking for an exit strategy.
The main take away is his frustrations center around his inability to staff up at the State Department without White House interference after Trump promised Tillerson he would have control over his department, a friend of Tillerson who is in touch regularly told CNN.
There are lots of frustrations which are apparent: personnel, Jared Kushner's outsized role, President Donald Trump undercutting him on several issues and Trump's general approach to foreign policy of inflicting chaos and confusion -- and the list goes on, the friend said, adding "it's becoming a death by one thousand little insults."
The friend said Tillerson is "buying time" and "evaluating all options."
Facing a myriad of challenges ranging from North Korea's nuclear threat to an evolving crisis in the Middle East, Tillerson's status as the administration's top diplomat has been called into question at times with Trump's senior adviser and son-in-law Jared Kushner taking the lead on several foreign policy initiatives.
The President has publicly contradicted Tillerson and others on several occasions, which has created some confusion about US priorities around the world as Trump continues to promote his "America First" philosophy which has drawn criticism from traditional allies.
Trump and Tillerson have pushed different messages on key issues including the crisis between Qatar and other Gulf neighbors, where the President tweeted that he hoped Qatar's isolation would hasten "the beginning of the end to the horror of terrorism!" Trump also used remarks in the Rose Garden
in June to reinforce his message.
In a sharp contrast to Trump's tone and messaging, Tillerson called on Gulf nations to de-escalate the crisis with Qatar, citing humanitarian, economic and military costs.
Despite their public differences it appeared as though Tillerson was beginning to emerge as a key voice on US foreign policy after he was the only American to join Trump
in a meeting with Russian President Vladimir Putin earlier this month.
But recent reports have raised new questions around his relationship with a White House that has not always been on the same page through the early months of Trump's presidency.
Multiple top officials at the State Department have told CNN they agree that Tillerson is not happy but indicated that he will likely stay in the post and give the situation some more time to improve.