Mylan has come under fire over the $600 price tag on its EpiPen, and the controversy has extended to pharmacy benefits managers like Express Scripts. Shares of Express Scripts dropped 6 percent Thursday.
Earlier in the day, Bresch discussed the repeated big price hikes, telling CNBC that intermediaries including wholesalers, retailers and pharmacy benefit managers add to the ultimate cost.
However, Express Scripts' Dr. Steve Miller said his company would love to see lower drug prices.
"If she wants to lower the price she can lower the price today," Miller said in a later interview with CNBC's "Closing Bell."
"We'd pass the savings that we take from the marketplace back to our plans."
While there have also been calls for overall transparency on drug pricing, Miller said the one area there should not be transparency is with his competitors. That way, a price war can be created that can drive down prices.
"Mylan has a monopoly and we've seen this time and time again. Because when companies have a monopoly for that temporary part of time, we actually can't have the free market work. We can't pit competition against each other and drive down the price, and she's capitalizing on it," he said.
Miller pointed out that in the last 18 months, Express Scripts' copays for EpiPens have risen less than 50 cents, from $73.03 to $73.50.
The pharmacy benefits manager trade group, the Pharmaceutical Care Management Association, also weighed in on the subject, saying Mylan was the latest drugmaker trying to reframe a pricing problem into a coverage problem.
"Blaming payers for these massive price hikes is a red herring and doesn't pass the laugh tests with policymakers," it said in a statement.
— CNBC's Dan Mangan contributed to this report.