Sen. Ted Cruz (R-Texas) on Sunday reiterated that the onus for defunding Obamacare remains on the House and offered a backup plan in case he and other GOP senators can't pass a budget that defunds the health care law.

"The House is the only body where Republicans have the majority, so the House has to lead on this," Cruz said on "Fox News Sunday." "And I view my job and (Sen.) Mike Lee's job as providing as much support, as much air cover as we can for the House to stand up and lead."

Cruz angered some House Republicans last week when, after House GOP leaders announced that they would vote on a budget that defunds Obamacare, Cruz expressed pessimism that he could stop the majority Senate Democrats from restoring that funding.

Cruz on Sunday suggested House Republicans should respond with another legislative maneuver.

"If Harry Reid kills this bill in this Senate, I think the House should hold its ground and begin passing smaller continuing resolutions one department at a time," Cruz said. "It should start with a continuing resolution focused on the military.

"Let's see if Harry Reid is willing to shut down the military just because he wants to force Obamacare on the American people."

As Roll Call noted Friday, the House has actually already passed a military funding bill.