A former orthopedic surgeon, Price, 62, is currently chairman of the House Budget Committee.
The official announcement is likely to be made soon.
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Trump had earlier said that he would repeal and replace Obamacare.
"The purpose of health reform should be to advance accessibility, affordability, quality, responsiveness, and innovation. None of these are improved by the Affordable Care Act," Price said in 2011.
"It's time for a better way to put patients, families, and doctors back in charge of medical decisions. President Obama and Democrats must put their pride and politics aside so that we can start over with a Better Way-a step-by-step plan to give every American access to quality, affordable health care," he said in October.
Price has himself come out with alternatives to Obama Care, which is similar to the vision of Trump.
"In choosing Price to fill a key Cabinet position, Trump will add to his team a staunchly conservative lawmaker close to the House Speaker. Price already is familiar with the budgetary process, the federal bureaucracy and the costs and mechanics both of changing the Affordable Care Act as well as revamping the Medicare entitlement program," The Washington Post reported.
Trump earlier nominated Indian-origin Nikky Haley as US envoy to the United Nations.
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