The committee sent a letter to Price asking that he "identify each usage of a private, non-commercial aircraft by a non-career official" by October 10.
Lawmakers are demanding information on political appointees' use of government planes for personal travel, as well as their use of private charters for official travel. "I will tell you I am not happy with it".
The exchange came days after the same publication reported that Price flew charter airplanes on more than two dozen occasions since last spring, costing taxpayers hundreds of thousands of dollars.
A former GOP congressman from Georgia, Price played a supporting role in the fruitless Republican effort to repeal Barack Obama's health care law, which has been another source of frustration for the president.
"I was looking into it, and I will look into it, and I will tell you personally I'm not happy about it", Trump responded when asked about Price's travel.
Price also traveled in June via chartered jet to Nashville, where he "spoke to a local health summit organized by a longtime friend" and had lunch with his son, Politico reported, citing an unnamed HHS official.
Uber wields new weapon in fight with London
Uber could remain on London's streets for more than a year - despite being banned by the capital's transport body last week. The company has 21 days to appeal the decision, and is allowed to continue working and driving throughout the process.
He has said he will halt private plane travel on official business while the internal review is being conducted.
Asked if he has confidence in Price, Johnson went to speak, then stopped himself and walked away.
"We'll see", Mr. Trump said.
Price, an ally of House Speaker Paul Ryan, is a past chairman of the House Budget Committee, where he was known as a frequent critic of wasteful spending.
Republicans on Capitol Hill have been slow to admonish Price, a former colleague, saying instead they are still trying to understand exactly what occurred.
"The American people deserve cabinet secretaries who place the public good above their own private interests", said Rep. Ruben Gallego (D-AZ), who said Price "should resign immediately". Yet one administration official noted that Trump has also been furious at Attorney General Jeff Sessions and several other top advisers who he has not fired.