The Democratic-led package to reform healthcare in the U.S. could win Republicans' votes next fall, Rep. Jeb HensarlingThomas (Jeb) Jeb HensarlingOvernight Finance: Lawmakers race to finalize omnibus | What we know about funding bill | White House on board | Fed raises rates for first time under Powell Power struggle threatens to sink bank legislation Despite Senate vote, Dodd-Frank reform is far from a done deal MORE (R-Texas) suggested Wednesday.

"When people see that they're going to lose the choice of their healthcare provider; that government agencies are going to ration their healthcare; that the quality of their healthcare will decrease; that they lose the choice of healthcare for their family, I think they will turn away from any big government, government bureaucrat, expensive healthcare plan," Hensarling said during an appearance on CNBC.

Hensarling said that Republicans would first have to convince voters of an alternative before they would see results at the polls.

"As Margaret Thatcher once said: First you win the debate, and then you win the vote," he said.

69 percent of Americans said they would favor a program that would increase the government's role in healthcare if it brought down costs and expanded coverage, according to a CNN/Opinion Research poll. 29 percent would oppose such a system. (That poll was conducted May 14-17, and had a 4.5 percent margin of error.)

Watch the Hensarling video below: