Upton, whose committee will play a key role in the GOP’s effort to roll back the law, said that he believes the House may be near the two-thirds majority required to override a presidential veto.

“If we pass this bill with a sizeable vote, and I think that we will, it will put enormous pressure on the Senate to do perhaps the same thing,” he said. “But then, after that, we’re going to go after this bill piece by piece.”