Rep. Tom Price got a privileged offer to buy a biomedical stock at a discount, the company’s officials said, contrary to his congressional testimony this month.

The Georgia Republican tapped by President Donald Trump to be secretary of the Department of Health and Human Services testified in his Senate confirmation hearings on Jan. 18 and 24 that the discounted shares he bought in Innate Immunotherapeutics Ltd., an Australian medical biotechnology company, “were available to every single individual that was an investor at...