Senate Majority Whip John Cornyn, R-Texas, said the debate over the Senate GOP's healthcare plan will be an "open process."

"This is going to be an open process where anybody on the Democratic or Republican side can offer an amendment, and it will get a vote," Cornyn said Sunday in an interview with NBC.

Senate Republicans released their plan to repeal and replace parts of Obamacare, called the Better Care Reconciliation Act, last month. But this week, GOP leaders revealed a new draft of the bill after conservative and centrist Republicans came out in opposition to the initial proposal.

Some centrist members have already criticized the updated measure for its cuts to Medicaid, and the plan has also been heavily scrutinized by more conservative lawmakers.

Senate Majority Leader Mitch McConnell was expected to hold a procedural vote on the bill early next week, but announced late Saturday he would defer consideration of the plan until Sen. John McCain, R-Ariz., returns to work.

McCain is spending this week in Arizona following a procedure to remove a blood clot in his left eye.

Cornyn told NBC that Republicans will "keep trying" to pass a plan to repeal and replace Obamacare if the current proposal fails, but conceded his party may have to start over.

"At some point, if Democrats won't participate in the process, then we're going to have to come up with a different plan," the Texas Republican said.

Though he didn't specify as to the timing of when a vote on the healthcare bill would take place, Cornyn said the Senate would move forward once McCain returns to Washington.

"I believe as soon as we have a full contingent of senators that we'll have that vote," he said. "It's important we do so."