On a late summer afternoon, Colin Allred, the Democratic challenger to U.S. Senator Ted Cruz of Texas, mingled his way through a barbershop in a historically Black neighborhood of Houston.

As he did so, a barber confessed that he had never heard of Mr. Allred, a Dallas congressman and former N.F.L. linebacker, until he was told the candidate would be coming by the shop. Neither had his client.

“Everybody knows Ted,” said the barber, Christopher Williams. “He’s not giving nothing up. It’s going to be a fight, and I love a good fight.”

In recent weeks, the race for U.S. Senate in Texas has grown increasingly competitive, with negative ads appearing with every channel flip and public polls showing a contest within single digits. Mr. Cruz, who is running for a third term, has been leading, but he attracts less support than former President Donald J. Trump does.

So what may have seemed like a sure thing for Republicans in the Republican-dominated state has some party officials in Texas concerned.

“The current reality is that Texas is too close for comfort,” Matt Mackowiak, head of the Travis County Republican Party, said in a letter to state party leaders.

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