Strategy to remove Secretary of State comes amid strained relations between Trump, top diplomat.
"You heard through the White House today that they have no personnel changes to announce," agency spokeswoman Heather Nauert told reporters.
"Secretary Tillerson enjoys this job, he has a lot of work to do," she said. "He does serve at the pleasure of the President. This is a job that he enjoys."
The call and comments follow media reports that said Kelly developed a plan to sack Tillerson and replace him with CIA Director Mike Pompeo.
The strategy to remove Tillerson comes amid strained relations between President Donald Trump and his top diplomat. Tillerson could be ousted in the coming weeks, the New York Times reported, citing anonymous senior administration officials.
But it is unclear if Trump has given his final sign-off on the proposal. If he does, Tillerson would be the latest in a series of senior officials to leave the administration following high-profile departures, including National Security Adviser Mike Flynn, Trump's chief strategist Steve Bannon and White House spokesman Sean Spicer.
Thursday's report is not the first time Tillerson has been rumored to be on his way out. In October, he dismissed claims he was planning an exit because of tensions with Trump. But the New York Times said he is intent on finishing the year before leaving.
Tillerson has been largely marginalized in the White House's decision-making process and he has faced widespread animosity within his department for inadequate staffing prompted by his planned overhaul of Foggy Bottom.
Tillerson has been publicly at odds with his boss during a number of major foreign policy developments -- urging a diplomatic solution to North Korea's nuclear and ballistic missile programs as Trump urged him not to waste his time, as well as in the Gulf crisis where Trump has directed rancor at Qatar as Tillerson seeks a mediated end.
He has also been a defender of the international community's nuclear deal with Iran -- an agreement Trump pledged to scrap on the campaign trail, and has continued to lambast after taking office.
If the plan to oust Tillerson is carried out, it would be the first time in more than 100 years that a top diplomat has left the post in such a short time without a change in administrations.
Following the report of the impending shakeup, Trump responded to questions on Tillerson's fate by saying "He's here. Rex is here."
White House spokeswoman Sarah Huckabee Sanders added there are not any personnel changes to announce at this time.
"Secretary Tillerson continues to lead the State Department and the entire cabinet is focused on completing this incredibly successful first year of President Trump's administration," she added in a statement distributed by the White House pool.
If Trump does give his final approval, Cotton and Pompeo would need to be confirmed by the Senate before taking their intended posts.
Pompeo has gained favor with Trump during his daily intelligence briefings with the president, oftentimes offering advice far beyond the CIA's mandate, but he has been criticized by his officers for being too political, according to the Times.