Credit ratings agencies are going to have to get used to a new regulatory regime in the wake of the Equifax consumer data hack, a top Washington regulator said Wednesday.

Consumer Financial Protection Bureau Director Richard Cordray said Equifax, as well as TransUnion and Experian, are going to be getting embedded regulators to ensure that similar breaches of private information don't happen again.

Companies "should welcome" the heightened level of scrutiny, Cordray told CNBC in a live interview on "Squawk on the Street."

"If they're going to restore public confidence in this marketplace and if they're going to create the kind of reforms necessary, they're going to have to recognize the old days of just doing what they want, being subject to lawsuits now and then, are over," he said. "There has to be a scheme of preventive monitoring in place. They're going to have to accept that, they're going to welcome that, they're going to have to be very forthcoming."

Fallout continues from the Equifax scandal, in which up to 140 million Americans — about 43 percent of the total population — had personal data exposed due to a breach discovered on July 29.

Multiple investigations have been opened on the matter, and CEO Richard Smith announced his retirement this week.

CNBC has reached out to the three companies for comment.

Cordray said the Equifax hack was "far beyond" what had happened at both Target and Home Depot and demanded a strong reaction.

This article was originally published by CNBC.