Price, 62, resigned yesterday, saying he regretted that "recent events have created a distraction" from the work of the health department.
The names of two Indian-Americans -- Seema Verma and Bobby Jindal -- have emerged as possible candidates.
He previously apologised after making 26 private flights since May.
Government officials, except those dealing with national security matters, are required to take commercial flights for work related travel. The total combined cost for Prices flights exceeded USD 1 million, according to media reports.
More From This Section
After Price's resignation, the US media immediately started speculating on the potential nominees for his replacement in the cabinet.
Most of the media outlets have placed Verma on the top of the list.
Currently administrator of the Centers for Medicare and Medicaid Services, Verma hails from Indiana, the home State of Vice President Mike Pence.
Trump so far has not indicated his choice.
Among the dozen odd potential candidates for the position, name of another Indian-American Bobby Jindal has also figured.
"But Jindal is steeped in health care," Politico reported.
"He was doing health policy academically and professionally before he entered politics," it said.
"Speculation about who would succeed Price permanently has focused on Seema Verma, who oversees Medicare, Medicaid and parts of the ACA. Shes a close ally of Vice President Pence. Former Louisiana Gov Bobby Jindal and Food and Drug Administration commissioner Scott Gottlieb are also seen as potential candidates," well-connected Axios reported.
Verma, who has been playing a key role in repeal and replace of Obamacare, has won accolades from Trump multiple times.
"The Trump administration and congressional Republicans might decide Seema Verma is too valuable exactly where she is," New York Magazine said.
"As for Bobby Jindal--- well, on paper he is perfect for the next HHS secretary," it said.
Other names floating around are Rock Scott, former Florida Governor; David Shulkin, Veterinary Affairs secretary; FDA Commissioner Scott Gottlieb; long-time House Republican Marsha Blackburn; Long-time HHS bureaucrat Don Wright; and Neurosurgeon Ben Carson.