Enzi, who is the chairman of the Senate's budget committee, proposes using the process known as reconciliation to undo much of the Affordable Care Act.
Reconciliation, which relates to budgetary matters, can be passed with a simple majority, as opposed to nonbudgetary bills, which can require 60 senators' votes to break a filibuster against them.
"Americans face skyrocketing premiums and soaring deductibles," Enzi said, according to a statement from his office obtained by CQRollCall.com.
"Insurers are withdrawing from markets across the country, leaving many families with fewer choices and less access to care than they had before — the opposite of what the law promised."
"Today, we take the first steps to repair the nation's broken health-care system, removing Washington from the equation and putting control back where it belongs: with patients, their families and doctors."