As Ted Cruz continued to hold the Senate floor, his senior GOP colleague from Texas was calling in to local radio stations expressing general support for the effort while noting a significant tactical disagreement.

“Ted’s been a great addition to the United States Senate,” Sen. John Cornyn said on KTSA. “He and I were dove hunting in Hondo this weekend together and we both agree that we need to defund and delay and defeat Obamacare to protect the American people.”

But Cornyn, the Republican whip, is in the camp along with Minority Leader Mitch McConnell, R-Ky., that plans to support limiting debate on the underlying House-passed continuing resolution, since that measure itself would block funding for the 2010 health care law that Cruz and other senators have spent all night speaking against on the Senate floor.

“I can’t see voting against that bill, which is what we’ve been asking for — for the last six months,” Cornyn said.

Cruz is portraying that vote as enabling a move by Democrats in favor of funding Obamacare, but Cornyn says that it is important to get Democrats from red states up for reelection in 2014 on the record supporting the law.

Cornyn noted the time crunch that the House would face over the weekend, particularly if senators are unable to reach a unanimous consent agreement to truncate the debate time on the continuing resolution that would keep the government running past Sept. 30.

“If we can’t defund it, I’m for delaying it, but here’s the challenge: The end of fiscal year is Monday,” Cornyn said. “Because of the Senate rules, it’s unlikely we’re going to finish this until maybe as late as Sunday, and so I don’t know how much time that Speaker Boehner and Majority Leader Cantor have to put together 218 votes in the House without seeing the government shut down.”

And Cornyn said a shutdown would hurt the GOP.

“As much as I loathe Obamacare, I don’t think a shutdown actually works in our best interests because the press and the mainstream media will inevitably side with Prsident Obama, and I don’t want to do anything to change the focus from President Obama and his failed policies, including Obamacare,” Cornyn said Wednesday morning.

The House has not yet signaled exactly how it will respond to the Senate. That’s no doubt in part because it is unclear precisely what the Senate will pass. Majority Leader Harry Reid, D-Nev., has announced his intent to change the expiration date in the stopgap measure to Nov. 15, and word came late Tuesday that Senate Democrats intend to eliminate a policy rider protecting biotech crops contained in the House version.

Sen. Jeff Merkley, D-Ore., issued a statement praising that expected move. He’s criticized the rider since it first appeared on a spending bill earlier this year.

“I applaud the hundreds of thousands of Americans who have worked hard to end this diabolical provision,” Merkley said.