Any move to replace Azar will be conditional on President Donald Trump deciding to move forward. The official stressed that nothing is imminent, but there are discussions in the White House about Azar's compensation.
As coronaviruses spread across the nation, the Trump administration has been criticized for its handling of the pandemic, namely by governors, on both sides of the aisle, in desperate need of critical supplies for its states, as the president has moved on to indicate that many states can reopen by May 1st.
In a statement Saturday night, White House deputy Judd Deere said: "The Department of Health and Human Services, headed by Secretary Azar, continues to lead a number of presidential priorities. Any speculation on personnel is unwarranted and a distraction from our response from the entire government to COVID-19. "
"Secretary Azar is busy responding to a global public health crisis and has no time for intrigue in the palace," HHS spokeswoman Caitlin Oakley said in response to the reports.
As a secretary, Azar has overseen several top management priorities, including attempts to weaken the Affordable Care Act, fight the opioid crisis and reduce the cost of prescription drugs.
Azar, who joined the Trump administration in January 2018, served as attorney general and deputy secretary of the agency under former President George W. Bush. He then joined Eli Lilly and became president of Lilly USA in 2012. He spent nearly nine years in the company, during which time they and other drug manufacturers hiked prices sharply.
He was appointed health and health secretary after his predecessor Tom Price resigned following the use of private aircraft for the government's business trips.
Azar, who was an economist and education lawyer, clerk for Supreme Court Justice Antonin Scalia in the early 1990s and later worked under Kenneth Starr, the independent lawyer in the Clinton Whitewater investigation that defended Trump during his impeachment trial.
This story has been updated with further developments.
CNN's Kristen Holmes, Tami Luhby and Maegan Vazquez contributed to this report.