The majority of House Freedom Caucus members will vote for a Republican health-care bill if changes offered by the White House are included in the legislation, the head of the conservative group of House Republicans said on Thursday.

U.S. Representative Mark Meadows said the group wants to see health insurance coverage waivers related to community rating protections with the exception of gender, essential health benefits and guaranteed issue.

"If those offers that were made over the last couple of days actually appear in the legislation, the majority, if not almost all of the Freedom Caucus, will vote for this bill," he said at a Politico news event.

An overhaul of the U.S. tax code could also be possible by August if U.S. House of Representative leaders are open to "serious debate," the head of the conservative House Republican group said.

Asked if lawmakers could pass tax reform before their summer break in August, Meadows told a Politico news event: "I do. I'm optimistic on tax reform if we do it differently than we did health care."

—CNBC contributed to this report.