Editor's note: This story has been updated throughout.
"Like Reagan said, never say never, but it's not something I'm spending a whole lot of time thinking about right now," McCaul told reporters Wednesday in Austin.
McCaul, the House Homeland Security Committee chairman, has been encouraged to take on Cruz following the Texas senator's controversial showing last month at the Republican National Convention, where Cruz declined to endorse GOP presidential nominee Donald Trump. Speaking with reporters at a book signing, McCaul acknowledged that "a lot of people" have urged him to challenge Cruz and said he was flattered by their support, but stressed he has nothing to do with the effort.
"It's, I think, been sort of organic, an effort to draft, if you will," said McCaul, who is advising Trump on national security. "But right now I'm really focused on my re-election to the Congress. I'm focused on advising the nominee to regain the White House and also maintaining a majority in the House of Representatives, which is critically important to the nation."
Asked whether he has been satisfied with Cruz's tenure in the Senate, McCaul said Cruz has "for the most part spent a lot of time running for president."
"I think he also represents the state of Texas in the Senate," McCaul told reporters. "I think that's an important job as well, and so I think the presidential campaign's over and it's time to — I think governance is important. I think in Washington getting things for the great people of Texas done is an important job."
Pressed on whether he was suggesting Cruz has not always been focused on Texas, McCaul replied, "Again, I think he's been focused on his ambition running for president."
A poll released Wednesday shows McCaul would face an uphill battle against Cruz in 2018. The Texas senator would beat McCaul by 32 points, according to the survey from the Democratic-leaning firm Public Policy Polling.
Cruz has nonetheless angered some Texas Republicans by withholding his support for Trump, his former bitter rival in the primaries. Asked how Cruz’s convention speech affected his standing in the state GOP, McCaul said he thinks “initially not well” but acknowledged it could depend on how Trump fares in November.
"Traditionally, you support your nominee for president, and so when I went to Cleveland, I gave a strong speech about Hillary Clinton and her devastating foreign policy, but also in the support of the nominee,” McCaul said. "I think that's an obligation that we have to support the nominee.”
McCaul spoke repeatedly of his role advising Trump on national security, saying at the book signing that it has not been an easy job. But he praised a recent anti-terrorism speech by Trump as “fairly effective” and “carefully crafted,” and he insisted he could not trust Clinton on national security.
Speaking with reporters, McCaul said he plans to use his advisory role to push Trump to change his posture toward Russia. Trump has been unusually friendly toward the country’s leader, Vladimir Putin.
“I think where he could use better advice would be with Mr. Putin and Russia,” McCaul said. "I don’t think Mr. Putin has our best interests at heart."