Expected Oct. 1 enrollment for health benefits from the Affordable Care Act might be delayed if Republican lawmakers in Washington are successful at defunding Obamacare.

House Speaker John Boehner, R-Ohio, said this week his plan to avoid a possible partial government shutdown as of Oct. 1 would also derail President Barack Obama’s health care law.

Republicans might have a hard time defunding the health care plan though, since Democrats as a whole have vowed to never let the health care law be delayed.

That isn’t stopping the plan to defund the Act by Republicans though.

On Thursday, U.S. Sen. Jeff Sessions, R-AL, cosponsored the Defund Obamacare Act of 2013. Sen. Ted Cruz, R-TX, introduced the act that would deny Obamacare federal funding for any implementation of enforcement activities, end any entitlements or benefits already put in place by Obamacare and rescind any unspent dollars that have already been appropriated.

“I have supported and will continue to support efforts to repeal, delay, defund, and block this unaffordable and invasive law,” Sessions said.

“Obamacare will not only make health care less affordable but will reduce the quality of health care and patient choice, increase unemployment, and surge the nation’s debt. The Defund Obamacare Act is the latest effort to protect American families from this unwise government takeover.”

The House has voted to repeal the Affordable Care Act more than 40 times. It is likely the measure by Sessions and Cruz will die in the majority-Democratic U.S. Senate.

The Associated Press contributed to this report.